An Ab Initio Approach to

the Inverse Problem-Based Design of

Photonic Bandgap Devices

Thesis by

John K. Au

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

2007

(Defended March 22, 2007)

ii

c© 2007

John K. Au

All Rights Reserved

iii

Acknowledgements

Finding an advisor is arguably one of the most impactful decisions in graduate school,

and I could not have chosen a better one than Hideo. If you ever talk to any of his

students, you will no doubt be told about the freedom we are given to try out ideas.

Rarely, though, do we follow up and explain that this freedom is not possible unless

he also has enough patience to allow these ideas to bear fruit, and the tolerance for

when they fail to do so. Perhaps this point is most significant to me as I may have

tried his patience more than any of my other labmates during my time here! Despite

my setbacks, whether they are research related or more personal in nature, he has

always remained unwaveringly positive, choosing instead to focus on ways to help me

move forward. I am very fortunate to have had an advisor who is as kind as he is

brilliant, which is tough to accomplish when he is a certified genius. Thank you for

finding ways to support me (for far longer than is perhaps deserved) so that I can

graduate.

I would also like to acknowledge many of the faculty members and other mentors

that have shaped how I think about science and research. In particular, I am indebted

to Dr. Cohen for teaching me to ask myself the question, “wouldn’t it be nice if . . . ?”

It did not get me over all my hurdles, but did help tremendously in leading me to

find the manageable yet still meaningful ones to climb over. I am also grateful to Dr.

Kimble for adopting me and the rest of MabuchiLab into his fold during our formative

years. One of the most memorable moments during my time here was the day I truly

understood the difference between quantum and non-classical. I was finally able to

appreciate and genuinely respect the passion and resolve he had always exhibited.

Finally, Dr. Herman has been an invaluable resource during the latter half of my

iv

time at Caltech. I have a much better appreciation for the process of research and

learning, and surely would have been too stubborn to learn that lesson without his

patient guidance through my most frustrating times. Outside of academics, he has

also taught me a great deal about life; from operating system upgrades to effective

parenting, and plenty more in between, for all of which I will always be grateful.

Many thanks to Parandeh, Marjie, Tara, Jim and Athena at the ISP for taking

care of all this paperwork for us international students. I am always amazed how

little I actually have to do despite nominally interacting with the government, and it

is definitely to your credit.

To survive graduate school, you definitely need friends who can commiserate with

you. My first couple years and all those classes would not have been the same without

George Paloczi, Tobias Kippenberg, Stephan Ichiriu and Will Green. Whether it was

trying to wrap up a problem set or deciding on a group to join, it was comforting

knowing that I was not alone. To the members of Professor Scherer’s Nanofab lab,

and especially Marko Loncar, thanks for treating me as one of your own. I only wish

I could have repaid you with better performance on the basketball court.

I had the great fortune of learning from excellent postdocs during my time here.

Thanks in particular to Andrew Doherty for spending a lot of time with me early on

teaching me everything from quantum trajectories to stochastic calculus (and on those

car rides back from ‘group meeting’ the secret to a good cup of French press coffee).

Jon Williams was instrumental in getting me up to speed on the physics of photonic

crystals, and I learned a great deal just generally about how to go about conducting

research from working with and observing him. To Luc Bouten, who remarkably

took an interest in my work, thanks for the encouraging words throughout my thesis

writing process. It meant more than you might have realized.

To the most intelligent set of ‘fools’ ever assembled, my fellow students in Mabuchi-

Lab: it has been an honor to have shared this part of our careers together. To the

young’uns Tony, Joe, Gopal, Nicole, Nathan and Orion: thanks for revitalizing the

foosball tradition. That foosball table has greatly increased both the quality and

quantity of my time at Caltech, though not necessarily in that order. May it live

v

on and continue serving MabuchiLab at Stanford, and my best of luck to you guys

in finding a regulation table. To Asa my officemate and politics liaison, thanks for

the interesting conversations outside of research, and for putting up with my work

area. Ramon deserves special mention for saving me in the eleventh hour by hacking

the CIT thesis style file. Thanks to Tim for sharing his experiences with seeking

an alternative career. Kevin ‘Employee Of The Month’ McHale deserves special ac-

knowledgement for his contributions to chapter 4 of this thesis, which turned out to

be instrumental in obtaining one of the key results of this work. I depart MabuchiLab

knowing my foosball moves could not be in better hands. A huge thank you to Sheri,

who keeps everything running smoothly despite having to put up with us juveniles.

Life definitely would not have been the same without the original crew, going way

back to in an era when group meeting had an Alias, and the a’s annihilated rather

than get annihilated. Good times . . . Ben, thanks for opening my eyes to what intense

passion for science is all about. Mike, my true Laker brother, how will I ever forget

chasing that factor of two with you? To Andy, for the numerous athletic activities

that helped keep me sane, and last but not least, Stockton, thanks especially for

helping me keep it real during the home stretch. I have many fond memories of our

years together. Thanks for being such an integral part of my grad school experience.

To the rest of my thesis defense committee Oskar Painter, Chiara Daraio and Axel

Scherer: Thanks for agreeing to be on my committee, and for the positive feedback

and insightful questions you had during the defense. To you, the reader. Even if you

are just reading the acknowledgements, I hope it has not been a waste of time.

My two little angels Charis and Akirin: you have brought such joy to me and

kept me balanced. Thanks for reminding me of the importance of wonderment and

curiosity.

And finally, my dearest wife Yuki, to properly thank you would more than double

the length of this thesis. I still cannot fathom how blessed I am to have you in my

life. Thanks for persevering with me, and just being with me throughout this journey.

This is our victory.

vi

Abstract

We present an ab initio treatment of the inverse photonic bandgap (or photonic crys-

tal) device design problem. Using first principles, we derive the two-dimensional

inverse Helmholtz equation that solves for the dielectric function that supports a

given electromagnetic field with the desired properties. We show that the problem is

ill-posed, meaning a solution often does not exist for the design problem. Our work

elucidates fundamental limits to any inverse problem based design approach for ar-

bitrary and optimal design of photonic devices. Despite these severe limitations, we

achieve remarkable success in two design problems of particular importance to atomic

physics applications, but also of general importance to the rest of the photonic com-

munity. As the first demonstration of our technique, we arbitrarily design the full

dispersion curve of a photonic crystal waveguide. Dispersion control is important

for maintaining the shape of pulses as they propagate along the waveguide. For our

second demonstration, we take a point defect photonic crystal cavity in the nominal

acceptor configuration (where the central defect has a lower index of refraction than

the bulk material) and force it into the donor configuration (where the defect has a

higher index of refraction than the bulk material), while requiring that the electro-

magnetic field maintain the properties of the acceptor mode. We were able to cross

over this threshold while retaining a 93.6% overlap with the original mode.

vii

Contents

Acknowledgements iii

Abstract vi

Contents vii

List of Figures xiv

List of Tables xv

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 2

I Mathematical Formalism 4

2 The Helmholtz Equation 5

2.1 Bulk Photonic Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Wave equation for E . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Wave equation for H . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 2D Plane Wave Expansion (PWE) Method . . . . . . . . . . . . . . . 8

2.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Supercell Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Point defect: cavity . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Line defect: waveguide . . . . . . . . . . . . . . . . . . . . . . 17

viii

2.4 Convergence Issues of the PWE Method . . . . . . . . . . . . . . . . 17

3 Inverse Problems 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.2 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Matrices as Linear Operators . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 A numerical example . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Singular value decomposition . . . . . . . . . . . . . . . . . . 29

3.3 Regularization and the L-curve . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 An alternate interpretation . . . . . . . . . . . . . . . . . . . . 34

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.1 Parameter estimation vs. design . . . . . . . . . . . . . . . . . 37

4 Convex Optimization 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Complex variables . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Convex Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Convexity conditions . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Gradient and Newton Methods . . . . . . . . . . . . . . . . . . . . . 50

4.4.1 Unconstrained optimization . . . . . . . . . . . . . . . . . . . 51

4.4.2 Incorporating constraints: barrier method . . . . . . . . . . . 58

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

II Device Design 63

5 Photonic Bandgap Devices: An Overview 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

ix

5.3 PBG and Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3.1 Waveguide dispersion design . . . . . . . . . . . . . . . . . . . 69

5.3.2 Large defect region cavity design . . . . . . . . . . . . . . . . 70

6 Inverting the Helmholtz Equation 73

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Inverse Problem Based Design . . . . . . . . . . . . . . . . . . . . . . 73

6.2.1 Genetic algorithms . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.2 Topology optimization methods . . . . . . . . . . . . . . . . . 77

6.2.3 Level set method . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2.4 Analytical inversion of waveguide modes . . . . . . . . . . . . 78

6.2.5 Our approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 Inverse Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . 80

6.3.1 Point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3.2 Line defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 Proof of Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4.1 Tikhonov solution . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.5 Simulating Design Errors . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.5.1 Convex optimization regularization (COR) . . . . . . . . . . . 88

7 From Inverse Problems to Device Design 91

7.1 3:1 Waveguide Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2 Residual Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.3 Inverse Problem Based Design Flow . . . . . . . . . . . . . . . . . . . 97

7.3.1 Valid eigenmode landscape . . . . . . . . . . . . . . . . . . . . 99

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.4.1 Comparing Tikhonov and Convex Optimization Regularization 103

8 Results 105

8.1 Waveguide Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.2 Enlarged Defect Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 106

x

8.2.1 The direct solution . . . . . . . . . . . . . . . . . . . . . . . . 109

8.3 Iterative Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3.1 Dispersion design . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3.2 Cavity design . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Appendices 118

A Sample Matlab Code to Illustrate Ill-Conditioning 119

B Barrier Functions of Complex Variables 121

C Fourier Transforms 123

C.1 Boundary Conditions and Fourier Space . . . . . . . . . . . . . . . . 124

C.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . 127

C.3 The Symmetry Problem . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.3.1 The underlying real-space function . . . . . . . . . . . . . . . 131

C.3.2 Even vs. odd . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

C.4 Fourier Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

D Detailed Errata for Geremia 2002 138

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

D.1.1 Relevant abstract of Geremia 2002 . . . . . . . . . . . . . . . 138

D.1.2 Typographical errors . . . . . . . . . . . . . . . . . . . . . . . 139

D.1.3 Proposed typographical errata . . . . . . . . . . . . . . . . . . 144

D.2 Mode Optimization Errors . . . . . . . . . . . . . . . . . . . . . . . . 146

D.2.1 Q factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

D.2.2 E field intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 147

D.2.3 Mode volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

D.3 Inversion Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xi

D.3.1 Correct derivation in the bulk mode basis . . . . . . . . . . . 153

D.3.2 Compare with PRE derivation . . . . . . . . . . . . . . . . . . 156

D.3.3 Solving the inverse equation . . . . . . . . . . . . . . . . . . . 157

D.4 Results Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Bibliography 160

xii

List of Figures

2.1 Geometry of 2D hexagonal lattice . . . . . . . . . . . . . . . . . . . . . 10

2.2 Geometry of reciprocal lattice of a 2D hexagonal lattice . . . . . . . . 11

2.3 Point defect cavity in supercell approximation . . . . . . . . . . . . . . 15

2.4 Line defect waveguide in supercell approximation . . . . . . . . . . . . 16

3.1 Stability of Hilbert matrix . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Stability of Hilbert matrix . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Linear transformation under the SVD . . . . . . . . . . . . . . . . . . 30

3.4 L-curve for the Hilbert operator inverse problem . . . . . . . . . . . . . 34

3.5 Spectrum of singular values for the Hilbert operator . . . . . . . . . . . 36

3.6 Spectrum of singular values for the inverse photonic problem . . . . . . 37

4.1 Graphical representation of test for convex function . . . . . . . . . . . 48

4.2 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Global underestimator of a convex function . . . . . . . . . . . . . . . 50

4.4 Backtracking line search . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Gradient method convergence comparison . . . . . . . . . . . . . . . . 54

4.6 Problems with the gradient method . . . . . . . . . . . . . . . . . . . . 55

4.7 Second-order approximation of objective function . . . . . . . . . . . . 56

4.8 Newton method convergence comparison . . . . . . . . . . . . . . . . . 57

4.9 The log barrier function . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.10 Modified objective function . . . . . . . . . . . . . . . . . . . . . . . . 60

4.11 Poor quadratic fit with log barrier . . . . . . . . . . . . . . . . . . . . 62

5.1 Photonic crystal waveguide . . . . . . . . . . . . . . . . . . . . . . . . 66

xiii

5.2 Comparing donor and acceptor modes . . . . . . . . . . . . . . . . . . 71

5.3 Fabrication constraints to acceptor mode . . . . . . . . . . . . . . . . . 71

6.1 Dielectric function for the h1 defect . . . . . . . . . . . . . . . . . . . . 84

6.2 Localized defect mode for the h1 defect . . . . . . . . . . . . . . . . . . 85

6.3 Proof of principle solution using QR factorization . . . . . . . . . . . . 86

6.4 Proof of principle solution using Tikhonov regularization . . . . . . . . 87

6.5 Proof of principle solution using Tikhonov regularization with initial

geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.6 Designing a noisy h1 mode using Tikhonov regularization . . . . . . . . 89

6.7 Designing a noisy h1 mode using convex optimization regularization . . 90

7.1 Target mode for the 3:1 splitter device . . . . . . . . . . . . . . . . . . 92

7.2 3:1 splitter device design output . . . . . . . . . . . . . . . . . . . . . . 93

7.3 L-curve for 3:1 Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.4 3:1 splitter device design output using L-curve corner . . . . . . . . . . 95

7.5 3:1 splitter device final design . . . . . . . . . . . . . . . . . . . . . . . 96

7.6 Output mode of 3:1 splitter device . . . . . . . . . . . . . . . . . . . . 97

7.7 Perturbing the dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.8 Perturbation to eigenmode . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.9 Sparsity of valid eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . 101

8.1 The PCW dispersion design problem . . . . . . . . . . . . . . . . . . . 106

8.2 Nominal h1 defect geometry . . . . . . . . . . . . . . . . . . . . . . . . 107

8.3 Cross-section of Gaussian broadened dielectric function . . . . . . . . . 108

8.4 Non-iterative result of waveguide design . . . . . . . . . . . . . . . . . 110

8.5 Comparing target and result for dispersion engineering . . . . . . . . . 112

8.6 Dispersion engineered waveguide design . . . . . . . . . . . . . . . . . 113

8.7 Comparing the target and the result for the cavity design . . . . . . . 115

8.8 Final enlarged defect cavity . . . . . . . . . . . . . . . . . . . . . . . . 116

8.9 Non-monotonic dependence of performance on iteration number . . . . 116

xiv

C.1 Fourier transform of a square pulse . . . . . . . . . . . . . . . . . . . . 126

C.2 Domain convention for the FFT . . . . . . . . . . . . . . . . . . . . . . 129

C.3 Shifting the origin in a 2D FFT . . . . . . . . . . . . . . . . . . . . . . 130

C.4 Underlying continuous functions corresponding to N FFT coefficients . 132

C.5 Even vs. Odd numbered FFTs . . . . . . . . . . . . . . . . . . . . . . 134

C.6 An arbitrary discontinuous real-space function in 2D . . . . . . . . . . 135

C.7 Comparing FFT coefficients in an even vs. odd grid . . . . . . . . . . . 136

C.8 Laurent’s Rule vs. Inverse Rule . . . . . . . . . . . . . . . . . . . . . . 137

D.1 Incorrect orthogonality condition . . . . . . . . . . . . . . . . . . . . . 152

xv

List of Tables

4.1 Performance of gradient method . . . . . . . . . . . . . . . . . . . . . 54

4.2 Performance of Newton’s method . . . . . . . . . . . . . . . . . . . . . 57

1

Chapter 1

Introduction

1.1 Overview

The field of photonic bandgap (PBG) materials, or photonic crystals, has expanded

tremendously over a relatively short period of time. These ‘semiconductors of light’

are artificially engineered materials that possess a periodicity in the index of re-

fraction, leading to bands of frequencies for which electromagnetic waves cannot

propagate in the material. Much effort has been poured into the field because of

the enormous potential for ground-breaking applications using these novel materi-

als. However, there have been many barriers towards taking full advantage of their

properties. The idealized PBG materials have a full three-dimensional periodicity,

but this has been difficult to fabricate, especially if intentional defects need to be

inserted at precise locations to add functionality to the material. Hence, most device

performances to date have yet to reach the promised potential, because they only use

the bandgap effect in two-dimensions.

The two-dimensional (2D) system is less ideal, because losses can occur in the

third dimension, so the coupling mechanism for the losses needs to be understood

for different in-plane configurations. This greatly increases the importance and com-

plexity of the design process. Since the field is relatively new (only 20 years), much

of the research has been performed by scientists using trial and error. Parameters of

the system are meticulously varied with the environment controlled, and the results

of those changes analyzed and studied. In true double-edged sword fashion, one is

2

fortunate in that one has enormous freedom on how to arrange the index of refraction,

so many designs are possible. Unfortunately it also means that it can be an endless

design challenge in the quest for better and better devices because of the vast number

of degrees of freedom.

We seek a better alternative to an exhaustive search technique for photonic crystal

device design, i.e., an algorithmic approach that removes the guesswork part of the

process. The basic idea behind an inverse problem approach is to formulate the

problem backwards. We start with the design goal we want accomplished, and then

work backwards to find the geometry that would have produced the intended effect.

Finally, our inverse problem design method is derived ab initio using Maxwell’s

equations. We do not make use of approximate models to the system we wish to

design. As such, for self-consistency we take great care in interpreting our results

accordingly, and the drawback is that the designs may not readily apply to real struc-

tures one can fabricate currently. However, we can make far more general statements

about the limitations and challenges of the design problem because our work is done

ab initio.

1.2 Organization of the Thesis

The work in this thesis is an amalgamation of several fields that traditionally have

minimal overlap. There has definitely been an increased interest in adapting inverse

problem techniques to the PBG problem in recent years, but given the amount of

research in each separate area, the degree of overlap is comparatively small. Convex

optimization methods is another established field of research that is being successfully

applied to many problems [1], but has yet to be relevant to the PBG community. As

such, this thesis is aimed at the practitioner in any one of these fields interested in

seeing how these come together to solve an engineering problem. Therefore, it is

written in a way that is accessible for an incoming graduate student in any of the

disciplines, although admittedly it does have more of a physics bias and assumes more

background in physics.

3

The thesis is divided into two parts. The first part develops the necessary math-

ematical formalism in the three areas. In chapter 2 we review the basic physics of

PBG materials and derive and work out the solution of the Helmholtz equation using

the plane wave basis. Chapter 3 discusses the idea of an inverse problem, and works

out a numerical example to illustrate what an inverse problem is and how to solve

these notoriously difficult problems. We conclude part I with a chapter on convex

optimization methods, which we will use as a specialized tool for solving photonic

inverse problems. Again, a numerical example is provided to help provide some basic

intuition into how the algorithm works.

Having provided the necessary mathematical background, we are then prepared to

shift our focus to the problem of device design. We begin part II with an overview of

PBG materials and devices in chapter 5. Readers familiar with PBG devices can safely

skip most of the chapter, although in section 5.3 we motivate the design problems that

we will tackle using our method. Chapter 3 reviews other inverse problem based design

methods in the literature, and highlights some advantages and disadvantages to our

approach. We derive ab initio the inverse Helmholtz equation, and perform a proof

of principle demonstration to conclude the chapter. We proceed to adapt the inverse

problem into a design methodology in chapter 7, revealing fundamental limitations

towards achieving optimal designs. Despite these difficulties, we demonstrate the

feasibility of a modified method that gives excellent results for our design goals that

cannot be obtained with other methods.

4

Part I

Mathematical Formalism

5

Chapter 2

The Helmholtz Equation

The underlying physics of photonic crystals follow Maxwell’s equations, and in this

chapter we derive from first principles the wave equation on which the work in this

thesis is based. There are many methods for solving the resulting partial differen-

tial equation (PDE), but the two most pervasive methods utilize either a time do-

main approach (e.g., finite difference time domain–FDTD)[2], or a frequency domain

approach[3], and each method has its own merits[4].

The time domain method, as the name implies, is well suited for studying dy-

namical properties of the fields such as pulse propagation, transmission/reflection

properties, estimate losses, etc. However, it is not as suitable for looking at resonant

behavior with a single frequency of interest. While it is possible to use FDTD to look

for resonant structures, one of the disadvantages is that computationally it is less ef-

ficient than the frequency domain approach, because you are looking at the system’s

entire response to a driving term. Another drawback of FDTD is that one must be

careful that the source term is not orthogonal to the mode of interest, in other words,

that there is sufficient coupling, or you risk not even finding the resonance. As the

linewidth of the cavity decreases, the required precision of the frequency increases,

which implies an increase in the number of timesteps required for the computation.

However, the most notable advantage of FDTD is that it scales more favorably [5]

than frequency domain methods.

As our group’s interest in photonic crystals began with PBG cavities, it was nat-

ural to adopt the frequency domain approach. We will see in this chapter that the

6

frequency domain problem using a plane wave basis reduces to a Hermitian eigen-

value problem, so there are many similarities to problems encountered in elementary

quantum mechanics. However, there are some known convergence issues with this

technique, and it is important that we draw attention to and highlight these in sec-

tion 2.4.

For a more general overview for studying the physics of photonic crystals, the

standard reference is the book by Joannopoulos et al. [6], or for a more mathematical

treatment, there is Sakoda’s book [7] as well.

Organization

We begin this chapter by deriving from Maxwell’s equations the wave equation for an

inhomogeneous medium. We show in section 2.2 the plane wave expansion method

for solving the Helmholtz equation to find photonic bandstructures. Adapting the

method for studying PBG materials with defect inclusions is reviewed in section 2.3.

Readers familiar with the method can safely omit those sections. We conclude in

section 2.4 with a discussion of the general convergence issues of the method. Even

though the information in that section is not new, there are too many references in

more current literature that seem unaware of the issues.

2.1 Bulk Photonic Crystal

Consider a position-dependent non-magnetic medium ε(r) with no charge or current

sources. We further restrict ourselves to consider only linear and scalar (i.e., isotropic)

materials. Maxwell’s equations for the various fields take on the following form:

∇ ·D(r, t) = 0 (2.1)

∇ ·B(r, t) = 0 (2.2)

∇×H(r, t) =∂D(r, t)

∂t(2.3)

∇×E(r, t) = −∂B(r, t)

∂t. (2.4)

7

Using the constitutive relations, we can eliminate D and B.

D(r, t) = ε(r)E(r, t) (2.5)

B(r, t) = µ0H(r, t) (2.6)

Note that unlike the standard derivation of the wave equation for a homogeneous

dielectric medium, we no longer have ∇ ·E = 0 from ∇ ·D = 0 because in general

∇ε(r) 6= 0. We can still work out the wave equations for E and H as follows.

2.1.1 Wave equation for E

We take the curl of eqn. (2.4) and combine with eqn. (2.6) to obtain

∇× (∇×E(r, t))

= −µ0∂

∂t∇×H(r, t) (2.7)

We substitute in eqn. (2.3) and combine with eqn. (2.5) to arrive at the wave

equation.

∇× (∇×E(r, t))

+ µ0ε(r)∂2E(r, t)

∂t2= 0 (2.8)

The standard frequency domain method then enforces harmonic time dependence of

the field to arrive at the Helmholtz equation for the electric field E.

E(r, t) = E(r)eiωt (2.9)

∴ ∇× (∇×E(r, t))

= εr(r)ω2

c2E(r), (2.10)

where εr(r) ≡ ε(r)ε0

. It turns out that the left side of the equation is not self-adjoint

(i.e. not Hermitian), so we are not guaranteed a complete basis [8]. Since the H

equation (to be derived below) is self-adjoint, we will work exclusively with that

equation instead, but for completeness we have presented the equation for E here.

Of course, once we have the H field, we can use eqn. (2.3) and (2.5) to obtain E.

8

2.1.2 Wave equation for H

The derivation for the H equation is similar to the E equation. We begin by com-

bining eqn. (2.3 and 2.5), take the curl and then incorporate eqn. (2.4 and2.6) to

obtain the following:

1

εr(r)∇×H(r, t) = ε0

∂E(r, t)

∂t(2.11)

∇×(

1

εr(r)∇×H(r, t)

)= ε0

∂

∂t∇×E(r, t) (2.12)

∇×(

1

εr(r)∇×H(r, t)

)= −µ0ε0

∂2H(r, t)

∂t2(2.13)

∇× (η(r)∇×H(r, t)

)= −µ0ε0

∂2H(r, t)

∂t2, (2.14)

where η(r) ≡ (εrr)−1 is the reciprocal of the relative permittivity function. To keep

the language from being overly cumbersome, we will refer to η simply as the ‘dielectric

function’ for the rest of this thesis.

Invoking harmonic time dependence, we arrive at our Helmholtz equation for the

magnetic field H .

∇× (η(r)∇×H(r)

)=

ω2

c2H(r) (2.15)

We also point out that in the derivation, we have not used two of Maxwell’s equa-

tions (Eqn. (2.1) and (2.2)), which means that a general solution to the Helmholtz

equation will not obey all of Maxwell’s equations. One must check or enforce these

conditions in order to have physically realizable fields.

2.2 2D Plane Wave Expansion (PWE) Method

Since equation (2.15) is self-adjoint we are therefore guaranteed a complete basis.

This implies that any spatially dependent function f(r) can be expressed as a linear

9

combination (i.e. superposition) of basis states.

f(r) =

∫a(k)φ(k, r)dk (2.16)

with

∫φ(k, r)φ(k′, r)dr = δ(k − k′), (2.17)

where δ(k − k′) is the Kronecker delta. The plane wave expansion method uses the

set of planes waves {exp(ik · r)} as the complete basis. One reason for choosing the

plane wave basis is that it is the eigenbasis in free space, so it is required for pseudo

Q factor considerations [9]. Another significant advantage to using plane waves is

that the vector fields will be divergence free (by construction), so all of Maxwell’s

equations (eqn. (2.1 and 2.2) in particular) are followed.

While it is certainly possible in principle to treat equation (2.15) vectorially in full

3 dimensions (3D) [10, 11], we will be making a 2D approximation for two reasons.

First, current fabrication technology has limited almost all PBG devices to take on

quasi-2D form, where the bandgap effect is utilized in only 2 dimensions and the

transverse dimension relies on total internal reflection for optical confinement. Second,

the computational resources required for a full 3D treatment of PBG devices using

the plane wave method is currently prohibitively expensive1. A final advantage of

the 2D treatment is that the TE (transverse electric) and TM (transverse magnetic)

polarizations are decoupled.

To be more precise, when we refer to a 2D approximation, we mean that the

dielectric function has full translational invariance in one dimension (say along the z

direction), so ∂η∂z

= 0. Under this configuration, the TE polarization has the electric

field vectors lying in the xy plane, and the magnetic field points along the z direction.

The TM polarization has the magnetic field in the xy plane and the electric field along

z.

1See [4] summarized in section 2.4 for an exception to this statement.

10

a1

a2

Figure 2.1: 2D hexagonal lattice in real space. The unit cell (dashed line) and theprimitive vectors a1 and a2 are shown.

2.2.1 Boundary conditions

For the case of photonic crystals, the dielectric function has translational symmetry

along the primitive lattice vectors {a1, a2} in the xy plane such that η(r) = η(r+R),

R ≡ m1a1 + m2a2, where {mi} are integers. For primarily historical reasons, we

choose as our canonical geometry a hexagonal (sometimes referred to as a triangular)

lattice of cylindrical air rods in semiconductor (see figures 2.1 and 2.2). The lattice

symmetry allows us to define Born von-Karman periodic boundary conditions (BCs)

for our PDE, as well as define a unit cell. Again, this naturally leads us to a description

in Fourier space, where because of the periodicity we also have a lattice (the reciprocal

lattice). The set of reciprocal lattice vectors {G} is defined by the following symmetry

requirement:

eiG · (r + R) = eiG · r (2.18)

∴ eiG · R = 1. (2.19)

We can leverage much of our intuition from solid state physics [12], and in fact, pho-

tonic crystals are often viewed as ‘semiconductors for light.’ In particular, Bloch’s

11

b1

b2

Figure 2.2: Reciprocal lattice in Fourier space of hexagonal lattice of figure 2.1

theorem applies (see chapter 8 in [12]), and we need only solve for the Bloch modes.

A final warning is that in invoking the Born von-Karman BCs, we have implicitly

assumed a photonic crystal of infinite extent. While this is a very standard approx-

imation, and it is a good approximation if the physical structure is large compared

to the lattice constant, it is important to be clear that the periodicity is meant to

extend to infinity.

Translational symmetry implies we can express:

η(r) =∑G

ηGeiG · r (2.20)

∴ ηG =1

Ac

∫

Ac

η(r)e−iG · rd2r (2.21)

H(r) =∑

G

hGeiG · r z (2.22)

∴ hG =1

Ac

∫

Ac

H(r) · z e−iG · rd2r, (2.23)

where Ac denotes the area of the unit cell, and in eqn. (2.22) we have chosen the

TE polarization. Eqn. (2.23) expresses the Fourier coefficient for some given Bloch

12

mode. In calculating band structures, the relationship is true though not helpful, as

we are trying to solve for the unknown Bloch mode.

We substitute into eqn. (2.15) the expressions in eqn. (2.20) and (2.22).

ω2

c2

∑

G

hGeiG · r z = ∇×

∑

G′ηG′eiG′ · r

∇×

∑

G

hGeiG · r z

=∑

G, G′ηG′hG∇× eiG′ · r∇× eiG · r z

=∑

G, G′ηG′hG∇× eiG′ · r(iG × z)eiG · r

=∑

G, G′ηG′hG∇× ei(G + G′) · r(iG × z)

=∑

G, G′ηG′hG

[i(G + G′)× (iG × z)

]ei(G + G′) · r

=∑

G, G′ηG′hG

[(G + G′) ·G]

ei(G + G′) · rz,

where in the last step we made use of the triple cross product identity a× (b× c) =

b(a · c) − c(a · b) and the fact that {G} lie in the xy plane. We then take the inner

product of both sides with a TE polarized plane wave by left multiplying and then

integrating:

1

Ac

∫

Ac

z e−iG′′ · r

ω2

c2

∑

G

hGeiG · r z =

∑

G, G′ηG′hG

[(G + G′) ·G]

ei(G + G′) · rz

d2r

ω2

c2

∑

G

hGδG, G′′ =∑

G, G′ηG′hG

[(G + G′) ·G]

δG + G′, G′′

We choose to collapse the delta function on the right with G′ = G′′ − G. Relabeling

13

the indices, we arrive at the Helmholtz equation in the plane wave basis.

∑

G′

[ηG − G′G · G′

]hG′ =

ω2

c2hG (2.24)

This is the starting point for bandstructure calculations of bulk (i.e. defect-free)

photonic crystals. So far we have only considered k-points that are coupled to the

q = 0 component, where {q} denotes the set of k-vectors that lie in the first Brillouin

zone (unit cell in reciprocal space). To complete the bandstructure to include the

other k-points, we simply use a different expansion of the field.

Hq(r) =∑

G

hq + Gei(q + G) · r z (2.25)

Here, q labels the plane wave that is used to modulate the Bloch function. In general,

the Bloch functions for different q will be different, as (following the same steps as in

the above derivation) we arrive at the following:

∑

G′

[ηG − G′(q + G) · (q + G′)

]h

n,q + G′ =ω2

n,q

c2h

n,q + G (2.26)

Written in this form, each q within the first Brillouin zone labels a unique eigenvalue

problem, and the n labels the band index of a particular mode. We do not need to

separately consider k’s outside the first Brillouin zone because all k’s that differ by a

translation of some G’ in {G} are coupled to each other. We use the band index to

keep track of the k’s in outer Brillouin zones.

Eigenfunctions that satisfy eqn. (2.26) are called ‘bulk modes’. The eigenvalues

give us the frequencies that are permitted within the material. When we solve a

series of these eigenvalue problems for q along high symmetry points, we obtain

the band diagram for the material. For a distribution of material with sufficient

dielectric contrast (e.g. low-index air rods in high-index semiconductors), we find

certain ranges where the frequencies are forbidden, which means that light at those

frequencies cannot propagate in the material, hence the term ‘bandgap.’

14

However, most useful PBG devices require that the symmetry be broken by in-

troducing defects in the lattice. Defects can be an omission of an air rod, or an air

rod of a different size or shape, or any other structure that causes a break in the

symmetry of the system. Structures with defects are studied in the PWE method

using the supercell approximation.

2.3 Supercell Treatment

The strategy here is to surround the defect(s) with enough layers of the bulk photonic

crystal that the modes of interest become well localized within the defect region.

We can then invoke the tight binding approximation and consider the defect plus

surrounding layers as a ‘unit cell,’ or a supercell. This implies a lattice of defects that

extend to infinity (see figures 2.3 and 2.4).

This is another standard approximation, and is valid if there is minimal interaction

between the artificial neighboring defect sites. This is analogous to the tight-binding

approximation in solid state physics. Note that for the line defect, the defects along

the x-axis should not be considered artifacts of the supercell method, since there is a

real translational symmetry in that direction (i.e. the waveguiding direction). When

solving for the modes of the waveguide, we need not be alarmed or concerned if the

mode is not localized in the x direction. However, that is not true in the y direction.

Finally, as with the bulk photonic crystal case, the translational symmetry of the real

waveguide does not extend to infinity even in the x direction.

2.3.1 Point defect: cavity

For the case of the point defect cavity, the extension to the Helmholtz equation is

trivial. Instead of the set {G}, we now have a new set of reciprocal lattice vectors

which we will refer to as {k}. We no longer have to worry about {q 6= 0} in the

Brillouin zone since they do not form a valid expansion. As we do not actually have

a supercell periodicity, physically we cannot support these longer wavelength plane

15

1

s22

Figure 2.3: Real space dielectric function of a hexagonal lattice of air cylinders em-bedded in an infinitely thick block of semiconductor. The point defect is formed byomission of the central air cylinder. In the supercell treatment, the entire devicemakes up the ‘unit cell’ (outlined in the dotted line) is artificially tiled to give pe-riodic boundary conditions. The original lattice points are shown as the faint reddots, but hold no particular special significance in the supercell treatment. The newsuperlattice points are the larger red dots, with the superlattice vectors s1 and s2

shown. The satellite defect structures are included but shown slightly faded out.

wave modulation of Bloch modes. The Helmholtz equation then simply sets q = 0

and substitute k’s for G’s.

In principle, the set {k} extends to infinity, although in practice we always trun-

cate at some finite bandwidth. There are some implications to truncation which are

often overlooked. We discuss these and other subtle issues with the discrete fourier

transform in appendix C. With a truncated basis, we can now obtain the Helmholtz

operator in matrix form.

∑

k′ηk − k′(k · k′)

hk′ =

ω2

c2hk (2.27)

Θkk′(η)

hk′ =ω2

c2hk (2.28)

16

s

s2

b1

b2

Figure 2.4: On the left side of the figure, we show the real space dielectric function ofa row defect in the supercell treatment. Small red dots correspond to original lattice,and the larger red dots the ‘basis vectors’ for the superlattice. On the right is apicture of the reciprocal lattice. The red dots show the reciprocal lattice of the bulkphotonic crystal, and the green dots show the reciprocal lattice of the new supercell.The shape of the original and supercell Brillouin zones are shown as well, with theoriginal one shown faded.

We have omitted the band index for clarity, since in general we will only be interested

in a small number of modes that lie within the bandgap. We denote Θkk′(η)

as the

Helmholtz operator, and the problem of finding the eigenmodes Hm(r) for some

given distribution of dielectric will be referred to as the ‘forward problem.’ The

superscript η makes it explicit that the operator depends on the chosen dielectric

function. We can form the Helmholtz operator once we have the Fourier coefficients

ηk, which can be obtained analytically or numerically. It is important to note here

that we set ηκ = 0 for κ = k − k’ that lie outside the truncation bandwidth. As

mentioned, appendix C will explore these issues in greater detail.

17

2.3.2 Line defect: waveguide

For the line defect waveguide, we also replace the G’s with k’s, but we must now

include the q’s that correspond to the propagation direction. Our forward problem

takes on the following form:

∑

k′ηk − k′

((q + k) · (q + k′)

) hq + k′ =

ω2q

c2hq + k (2.29)

Θkk′(q,η)

hq + k′ =ω2q

c2hq + k (2.30)

again, omitting the band indices and making the dependence on η explicit. The

dispersion relation ω(q) of the waveguide can be found by solving the forward problem

for several q’s along the propagation direction, and identifying the waveguiding mode

of interest whose frequency lies within the bandgap.

2.4 Convergence Issues of the PWE Method

In the opening section of this chapter, we alluded to some known convergence issues

with the PWE method. From working out the Helmholtz equation in the plane wave

basis, the connection is clear between the plane wave method and Fourier analysis so

it is not surprising that any difficulties in Fourier analysis (see appendix C) will lead

to difficulties here. However, as it applies to solving the photonic bands problem,

these issues were studied as early as 1992 by Sozuer, Haus and Inguva [13]. This was

in an era when the field of photonic crystals was still in its infancy, and tremendous

amount of effort went into accurate calculations of the band structure in search of

true band gaps. In an effort to correct a common misconception, they wrote the

following:

It is clear that, just because increasing N does not produce visible differ-

ences in the resulting band structure, one has not necessarily converged

to the ‘true’ values. In this case, it is merely an indication of the slow

18

convergence of the Fourier series.

Here, N refers to the number of terms in the summation. Unfortunately, most con-

vergence analysis on the PWE method fails to address the problem, and even research

published over a decade later [14, 15, 16] still fails to grasp the difference between

convergence and accuracy. After all, what is the significance of ‘rapid convergence’

of the calculation if it does so to a wrong value? The problem is, of course, the

slow convergence of the underlying dielectric function that the finite Fourier series is

supposed to model. Additional terms in the series do not change the model enough

so it only appears as though the calculation has converged.

In the literature, there was also some discrepancy as to how to treat the η = ε−1

term in the Helmholtz equation (eqn. (2.15)). While some have treated the Helmholtz

operator as we have, others [11] expand εr(r) in the plane wave basis, and then invert

the matrix instead. For an infinite Fourier series, the matrices ηk,k′ and εk,k′ would

be each other’s inverse, but that is no longer true once we truncate. It appeared that

the convergence was more rapid (i.e. used a fewer number of plane waves) using the

matrix inversion treatment [17], thus justifying the computationally intensive matrix

inversion, but bear in mind the caveat about convergence from above. Of course, in

the early 1990’s, there were more constraints on CPU and memory resources than

there are today. This issue is addressed nicely by Li in 1996 [18], and we will review

his work in section C.4. In the end, the point is moot, since Steven Johnson et

al. found a way [4] to implement the plane wave method without explicitly forming

either matrix, and have since made their software, the MIT Photonic Bands (MPB)

package, freely available.

There are three key ideas to their approach. First, rather than solving the eigen-

value problem explicitly by diagonalization (computational work required O(N3)),

they use an iterative eigensolver. Secondly, they noticed that in the Helmholtz equa-

tion, the curl operator is diagonal in Fourier space, while the division by ε is diagonal

in real space, so they make use of the Fast Fourier Transform (FFT) algorithm to

transform to the appropriate basis so they do not actually store the entire N ×N op-

erator. They were thus able to reduce their storage requirement from O(N2) to O(N).

19

This allowed them to use a much higher number of plane waves than otherwise prac-

tically achievable. However, representation of discontinuities in a Fourier basis still

poses a problem. The final ingredient is an averaging technique that smoothes out the

discretized elements that encloses a discontinuity. It makes use of effective medium

theory and the grid elements containing the discontinuities are assigned a dielectric

tensor instead of a scalar. Therefore, they have replaced the sharp scalar dielectric

discontinuity with a smoothed dielectric tensor, making the Fourier representation

less objectionable.

We have chosen to highlight some of the key contributions of that work here for

two reasons. The first reason is that we were unable to take advantage of their ideas

in the inverse problem, so we still face the same convergence issues described in [13].

It should become clear when we formulate the inverse Helmholtz problem in chapter

6 why we could not capitalize on their wisdom. The other reason is that though

their work is often cited, there still appears to be confusion about the validity of the

PWE method, especially the key components to making the method work. Other

more recent articles on PWE [14, 16, 15, 19] either omit the reference entirely, or if it

does cite it, the work demonstrates a complete lack of appreciation for the results by

Johnson et al.. A recent article (2006) on these ‘fast Fourier factorization methods’

[20] still have not caught on to the fact that their method is at best equivalent if not

inferior to the tensorial averaging in the Johnson reference, except that they do not

even realize the N log N scaling, requiring O(N2) storage. For the reader interested

in performing these computations, it is important to understand the significance of

Johnson’s work and to evaluate other PWE method research in light of their results.

20

Chapter 3

Inverse Problems

Mathematicians often cannot help but cringe when physicists are doing math, and

even more so whenever physicists claim to be ‘rigorous’ with their math. This chapter

is not written to satisfy the mathematicians for two reasons. First, I am not a

mathematician, so I am quite certain that they will cringe despite my best efforts.

More importantly though, this is meant to be accessible for engineers and physicists,

and often what mathematicians consider ‘special cases’ are the only ones we happen

to care about. So with apologies to any mathematicians reading this, the goals of

this chapter are threefold: First, we want to help the reader develop an appreciation

for what inverse problems are and what makes them difficult. Second, we want to

introduce the specialized tools that are used to solve these inverse problems. Finally,

we bring the focus back to our particular application, and fine tune the ideas developed

for the purpose of photonic device design.

There are many excellent references on inverse problems. A standard reference is

the textbook by Engl [21] which gives a thorough overview of the subject, but the

mathematics is quite formal. A very nice introduction to the subject for physicists

can be found in a series of lecture notes by Sze Tan and Colin Fox at the University

of Auckland [22]. The work by Per Christian Hansen is more focused on discrete

and finite dimensional problems, and hence particularly suitable to our application.

He has also written a package of matlab functions for inverse problems available for

download as well [23]. The ideas presented in this chapter are mostly taken from his

work, although the discussion of the role of noise in distinguishing between a forward

21

and inverse problem has to our knowledge not been articulated elsewhere. Arnold

Neumaier also provides a concise treatment similar to Hansen’s, but bridges the gap

to the infinite dimensional treatment of inverse problems with more mathematical

rigor [24].

We begin the chapter by attempting to define what an inverse problem is through

some examples of simple physical problems. We introduce the concept of an ill-posed

problem to distinguish between the forward or direct problem vs. the inverse problem.

In section 3.2, we restrict our discussion to finite dimensional linear operators, allow-

ing us to illustrate the pathologies of inverse problems in the linear algebra formalism.

A numerical example is provided to help illustrate the effects of ill-conditioning. We

make use of the singular value decomposition (SVD) to explain why standard tech-

niques will fail to solve inverse problems. The SVD also allows us to utilize the

condition number as a quantifying metric for how ill-posed a particular problem is.

In section 3.3 we introduce regularization as a tool for solving inverse problems. We

conclude with a glimpse of the difficulties we expect to encounter for the purpose of

PBG device design.

3.1 Introduction

At first glance, the meaning of the term ‘inverse problem’ seems obvious. It is the

complement of some other problem, one that presumably preceded the inverse prob-

lem, and is more well known. To a physicist though, such a ‘definition’ is rather

unsavory, for if that were the case, then the distinction between a forward problem

and an inverse problem seems rather arbitrary. Our obsession with symmetries in

natural laws lead us naturally to wonder why one problem formulation is more ‘priv-

ileged’ than the other. A good example of this that we have already encountered

in this thesis is the Fourier transform and the inverse Fourier transform. The two

operations are simply labeled that way by convention, and nothing would have been

lost had we reversed the labels. It becomes a question of semantics, rather than a

matter of any fundamental significance. By the end of this chapter, we will see that

22

the inverse Fourier transform in fact does not fit our definition of an inverse problem.

The distinction is in reality more than just semantics or there would not be an

entire journal devoted to inverse problems. One’s first exposure to inverse problems is

typically accompanied by some claim that inverse problems are difficult to solve, with

the implication being that it is more difficult than the associated forward problem.

We give a more formal definition in the next section, but first, we review a few

well-known examples of inverse problems to develop some intuition.

3.1.1 Examples

Our first example is found in medical imaging, such as computerized tomography

(CT) scans. The forward problem is a form of scattering or diffraction problem, such

that for some radiation incident upon a given material distribution, we determine

the scattered radiation in the far field. For medical applications, the goal is to non-

invasively determine the internal structure of a patient’s body. This is accomplished

by measuring the scattered field at various angles given some incident radiation, and

solving the inverse scattering problem for the scatterer distribution. A related inverse

problem is found in geophysics, where the internal structure of the earth is determined

based on surface measurements of seismic waves.

Another example is in image processing, or image restoration, where the ideal

image must pass through non-ideal optics, leading to blurring and other distortion

to the captured image. The forward problem of blurring is typically modeled as

a convolution of the original image io(x) with a point spread function h(x). Sharp

features are smeared out, leading to a loss of resolution. Formally, our captured image

ic(x) becomes:

ic(x) =

∫io(ξ)h(x− ξ)dξ, or (3.1)

Ic(k) = Io(k)H(k) (3.2)

The inverse problem becomes a deconvolution, which can be performed as a simple di-

vision of the captured image with the point spread function in their respective Fourier

23

representations. One can determine h(x) by characterizing the optical elements care-

fully. It turns out that the image cannot be reconstructed with a straightforward

application of the deconvolution theorem. Section 1.6 in [22] provides a nice pictorial

example of this problem.

A final example is the heat conduction problem. The forward problem is of course

a standard undergraduate-level problem, and can be described by some variant of the

following:

∂u(x, t)

∂t=

∂2u(x, t)

∂x2, x ∈ [0, π], t ≥ 0, (3.3)

where u(x, 0) = f(x), (3.4)

u(0, t) = u(π, t) = 0. (3.5)

This is solved in the usual way by separation of variables and then an eigenfunc-

tion expansion for the spatial dependence, with the set of normalized sine functions

{φn(x)} forming a complete orthonormal set. Expressing the initial distribution in

terms of a superposition of the eigenfunctions f(x) =∑

n cnφn(x), we obtain the heat

distribution u(x, t) as

u(x, t) =∑

n

cne−n2tφn(x). (3.6)

There is often some remark about our inability to solve the backwards heat conduc-

tion problem, namely given some final distribution u(x, tf ), we generally cannot go

backwards in time and deduce an initial distribution. Typically this is attributed to

the exponential factor, and we see that it blows up if we go backwards in time.

Based on these examples, we can make some observations that will prove to be

helpful. First, the heat conduction example makes explicit a common theme among

the examples provided: that of cause and effect. Whereas forward problems in physics

tend to study the unknown effects of known causes (in order to derive a model for pre-

dicting the effects), inverse problems seek the unknown cause of measured effects. The

backwards heat conduction equation makes this transparent because of the explicit

time dependence, but the rest of the examples all seek an explanation of some final

24

observed phenomenon, given some well-characterized, presumably accurate model of

the forward problem.

The other observation is that at least in some of these forward problems, there ap-

pears to be a ‘smoothing’ process. For example, in the heat conduction and the blur-

ring examples, features present in the initial condition seems to get lost or smoothed

out as the state evolves forward. Often, we arrive at steady state solutions of dy-

namical systems, therefore independent of initial conditions: The system forgets or

loses information about the initial state. In such a case, we can certainly see just by

physical principles alone why an inverse problem would be ‘difficult.’ A more precise

way to look at this might be how ‘solvable’ a given problem is, which leads to the

notion of well-posed and ill-posed problems proposed by Hadamard.

3.1.2 Well-posedness

Hadamard proposed three properties that a problem must possess in order to be

classified as well-posed [21]:

1. For all admissible data, a solution exists.

2. For all admissible data, the solution is unique.

3. The solution depends continuously on the data.

What constitutes ‘admissible data,’ ‘solution’ and ‘continuous’ will of course depend

on the nature of a specific problem. For our purposes, considering only finite dimen-

sional linear operators, we can think of data and solution as the input and output

vectors of some linear transformation. The first two properties seem rather obvi-

ous, as it is not much of a linear mapping if we, for some given input, cannot get

an output, or get multiple or non-unique outputs. The third property is a question

of stability, requiring that small changes to the input does not produce arbitrarily

large changes to the output. A problem that lacks any one of these properties is by

definition ill-posed.

25

We can apply our intuition to the heat conduction example and readily see that

indeed a solution to the inverse problem does not always exist. A simple example

is if our final field distribution corresponds to a state with minimal entropy. Since

entropy must increase with time, we know that there is no way to go backwards in

time, since we are ‘starting’ in a minimum entropy state. As for the uniqueness of the

solution to the inverse problem, we already addressed the problem of the steady state

fields, which means any initial state would reach the same steady state. The final

property of stability relates to the smoothing behavior of the forward problem. If we

perturb the initial conditions by a small amount, the perturbations will be smoothed

out over time to yield similar output fields. By extension then, small perturbations

at the final time must have corresponded to large changes in the initial condition.

We observed this effect quantified by the exponential term in the heat conduction

equation. We now express these ideas in a more formal mathematical footing in the

context of finite dimensional linear operators which can be represented by matrices.

3.2 Matrices as Linear Operators

3.2.1 A numerical example

We first provide a numerical example to give a concrete illustration of the ideas

presented in the previous section. The matlab code used to generate this example is

included in appendix A. Consider the N ×N Hilbert matrix.

A(i, j) =1

i + j − 1(3.7)

with Axin = xout (3.8)

26

For this example, we will choose N = 5, and choose a relatively simple xin.

A =

1 12

13

14

15

12

13

14

15

16

13

14

15

16

17

14

15

16

17

18

15

16

17

18

19

, xin =

1

1

1

1

1

(3.9)

Evaluating xout gives:

xout =

1 12

13

14

15

12

13

14

15

16

13

14

15

16

17

14

15

16

17

18

15

16

17

18

19

1

1

1

1

1

=

2.2833

1.4500

1.0929

0.8845

0.7456

. (3.10)

Clearly, for any xin, we can evaluate a unique xout. Therefore the first two Hadamard

conditions are satisfied. To test the stability condition, we can define an additive

noise vector n that is sufficiently small to form x′in. Evaluating x′out gives:

x′out =

1 12

13

14

15

12

13

14

15

16

13

14

15

16

17

14

15

16

17

18

15

16

17

18

19

0.9982

0.9992

0.9997

1.0003

1.0010

=

2.2813

1.4490

1.0922

0.8840

0.7452

. (3.11)

We see that x′out is close to the nominal solution xout. Formally we can define the

relative magnitude of the input and output error to provide a measure of the stability

27

0 0.2 0.4 0.6 0.8 1 1.2 1.40

50

100

150

200

250

300

350

400

450

Stability

Figure 3.1: Distribution of stability values for the Hilbert matrix operator

S:

ein =|x′in − xin||xin| (3.12)

eout =|x′out − xout|

|xout| (3.13)

S ≡ eout

ein

= 0.7846 (3.14)

where | · | denotes the 2-norm (i.e., |x| = (∑

i x2i )

1/2). We repeat this with 10, 000

different noise vectors and show the distribution of S in figure 3.1. Most of the values

fall between 0 and 1, with the maximum value of about 1.1. Therefore, we see that

this problem is stable against perturbations to the input vector, i.e., errors remain

small.

We now look at the ‘reverse’ problem of finding xin given xout. We will look at

the stability again, but this time, we add the noise to the nominal xout. We solve for

28

x′in = A−1xout. We use matlab to find the inverse of A.

A−1 =

25 −300 1050 −1400 630

−300 4800 −18900 26880 −12600

1050 −18900 79380 −117600 56700

−1400 26880 −117600 179200 −88200

630 −12600 56700 −88200 44100

(3.15)

Again, using 10, 000 different noise vectors, we obtain the distribution of S for this

reverse problem (as shown in 3.2). Notice the x-axis is scaled by 105, meaning the

relative error is greatly amplified. To illustrate, suppose we rounded xout to 3 decimal

places and then evaluated x′in.

x′out =[

2.283 1.450 1.093 0.885 0.746]T

(3.16)

x′in = A−1x′out =[

2.105 −20.28 94.29 −141.4 71.19]T

(3.17)

In the reverse problem, we cannot even tolerate rounding errors as x′in bears no

resemblance to xin at all. Therefore, this problem fails to satisfy Hadamard’s third

condition and is therefore ill-posed. Because of the ill-posedness, this reverse problem

is the one that is defined to be the inverse problem. Therefore, it is not simply a

question of semantics, but there are fundamental distinctions between a forward and

its inverse problem. Even if we had first defined an operator B = A−1 and went

through this same analysis, we would still conclude that B is the ‘inverse problem,’

and B−1 is the ‘forward problem,’ objectively based on the stability criterion.

In chapter 6, when we derive the inverse Helmholtz equation, we will encounter a

more severe manifestation of this problem, where we cannot (even without the additive

noise) recover the input with the computed output. However, if we understand the

analysis in this chapter, it will no longer be surprising when we get to chapter 6. To

understand the origins of this ill-conditioning, we need to take a closer look at the

properties of a general linear operator A in terms of the singular value decomposition.

29

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 105

0

50

100

150

200

250

300

350

Stability

Figure 3.2: Distribution of stability values for the inverse Hilbert matrix operator.Notice the scale on the x-axis is in increments of 105.

3.2.2 Singular value decomposition

Consider a general linear transformation (or linear mapping) A : Cn → Cm, such that

Ax = b (3.18)

with A an m× n matrix, x ∈ Cn and b ∈ Cm.

Any matrix A can be decomposed by the singular value decomposition (SVD)

such that

A = UΣV †, (3.19)

where the m×m matrix U and the n× n matrix V are unitary, and Σ is an m× n

matrix whose only non-zero elements are along the diagonal with {σi ≥ σi+1 ≥ 0}called the singular values. The columns of U and V are known as the left {ui} and

right {vi} singular vectors. This is a generalization of the eigenvalue decomposition.

In fact (although one would not actually do so in practice), one can get the SVD by

performing an eigenvalue decomposition of AA† and A†A. The eigenvectors of AA†

and A†A are the left and right singular vectors of A respectively, and the eigenvalues

are the singular values squared. Since both AA† and A†A are Hermitian and positive

semi-definite, we are guaranteed real non-negative eigenvalues, thus ensuring σ ≥ 0.

30

v2

v1

u2

u1

σ2α

2

σ1α

1

α2

α1

α2α

1

xAx

Figure 3.3: The vector x to be transformed is decomposed into v1 and v2 and thenmapped onto corresponding u1 and u2, with each component stretched or compressedby the respective σ. In this example, σ2 < 1.

Having obtained the SVD of A, we can write down the linear mapping in a more

suggestive form:

Ax = UΣ(V †x) (3.20)

Ax =

min(m,n)∑i=1

(v†ix

)σiui, (3.21)

where for clarity we have written the vectors in boldface. Looking at eqn. (3.21), we

see that any linear mapping can be viewed as a transformation of the input vector x

into the right singular vector basis, then stretching each component by the associated

singular value, and finally mapping these components to the corresponding left singu-

lar vectors. A pictorial representation for a simple 2D mapping is shown in figure 3.3.

For special cases of A that has an eigenvalue decomposition (i.e. diagonalizable), the

left singular vectors are the same as the right singular vectors, so in the diagonalized

basis, the linear transformation is particularly simple (just stretch each component

by the eigenvalues; this is, of course, why we prefer to work in a diagonalized basis).

The singular values of A play an important role since they determine how much gain

is in a particular component of the linear map. For the time being, let us consider the

problem of finding b (given A and x) to be the forward problem, while the problem

of finding x (given that Ax produces b) is the inverse problem.

31

Revisiting Hadamard’s conditions then, in the forward problem, the first two

conditions are automatically satisfied if we can express the problem in this form. As

for the third condition, we can think of it as requiring reasonable gains (i.e. not too

large) for the system. Stability can also be achieved if random perturbations are

spectrally decomposed to singular vectors that have relatively small singular values.

In other words, singular vectors associated with small singular values should look like

noise. For physical systems the relevant physics are embodied by the linear operator

A.

If we now attempt to solve the inverse problem, we need to do the following:

Ax = b (3.22)

x = A−1b (3.23)

= V Σ−1U †b (3.24)

=

min(m,n)∑i

(1

σi

u†ib)

vi. (3.25)

In eqn. (3.25) above, we have expressed the inverse of A using the SVD expansion.

Even when A is singular (i.e. not strictly invertible), the expression can be used and

interpreted as a generalized inverse or ‘pseudo-inverse,’ although there are of course

limitations associated with a singular A. Now A is obviously singular when m 6= n,

but even when A is square it can still be singular if σi = 0. Singularity of A implies

that A does not have full rank, i.e. A is rank deficient, or A has a non-trivial nullspace:

For σi = 0, (3.26)

Avi = 0 (3.27)

∴ A(x + αvi) = b, and furthermore, (3.28)

@ y | Ay = βui (3.29)

Eqn. (3.28) shows that we fail the uniqueness test, and eqn. (3.29) shows that we

fail the existence test for Hadamard’s condition for well-posedness. In most physical

32

problems, the singular values may not be identically zero, as it would be impractical

to numerically evaluate them to that level of precision. Based on the importance of

the singular value spectrum though, we can define a condition number :

C ≡ max σ

min σ(3.30)

As C becomes larger, the problem becomes more ill-conditioned, and for a strictly

singular matrix, C → ∞. Even though we now have this quantity defined, the

boundary between what is considered well-conditioned and poor-conditioned is not a

sharp one. A generally acceptable figure is C ≤ 103.

We now return to our numerical example of the Hilbert matrix. The singular val-

ues are {1.567, 0.2085, 0.0114, 0.0003, 0.000003}. The condition number is 4.766×105,

so as suspected, the problem is ill-conditioned. Specifically, let us examine eqn. (3.25),

especially the factor σ−1i . In the forward problem, the small singular values damp

out the contributions from the additive noise. In the inverse problem, however, they

become an amplification for the noise components, drowning out the original sig-

nal xin. This amplification picture is consistent with our result above, as we found

|A−1x′out = x′in| >> |xin|. If a problem is ill-conditioned, any standard matrix inver-

sion algorithm will fail to recover the desired solution xin. Having understood the

origins of the difficulties, we can now discuss strategies for overcoming these difficul-

ties.

3.3 Regularization and the L-curve

The specialized technique that is used to solve inverse problems is called regulariza-

tion. There are many regularization schemes that have been developed, and Engl’s

text is a good starting point. Here, we will only discuss the most common regulariza-

tion scheme, known as Tikhonov regularization, that works well in many situations.

First, we must address the three properties of ill-posedness. In a way, they are

related, because any time we have a singular (or near-singular as defined by C)

33

matrix, any of the three can occur. The lack of an existence theorem is overcome

by minimizing the residual |Ax′in− x′out| in usual inverse problem applications, and is

not considered too serious. One must give up on the notion of an exact solution to

inverse problems. Rather, we just try to reconstruct a ‘sensible’ solution that satisfies

our given equation ‘well enough.’ We will comment further on this issue in the final

section of this chapter.

Non-uniqueness is considered much more serious. In our numerical example, hav-

ing given up the notion of an exact solution, we know that |Axin − x′out| would have

been nonzero but small. In fact, it is exactly the norm of the small noise term added

to xout. The problem becomes how to pick out the nice solution among all the many

that would still give reasonably small residual norms. We observed at the end of the

last section that the small singular values lead to large noise amplification. We note

also that these bad solutions do tend to blow up and have large norms, much larger

than the desired solution. Therefore, one strategy would be to restrict the size of

the solution. This additional constraint allows us to choose systematically a unique

solution that at least allows a ‘sufficiently small’ residual. Rather than minimizing

only the residual, we can include the solution norm as well to formulate the following

regularized inverse problem:

x(λ)out = min

xout

{|Axout − x′in|2 − λ|xout|2}

(3.31)

The scalar λ is known as the regularization parameter, and determines the relative

weight between minimizing the residual norm versus the solution norm. The standard

graphical tool that assists in choosing the regularization parameter is the L-curve. The

L-curve plots the solution norm on the y-axis and the solution norm on the x-axis for

a series of λ’s. The L-curve for our numerical example is shown in figure 3.4, and it

takes on its name because of the characteristic L shape. The corner is usually taken as

the appropriate regularization parameter that indicates optimal compromise between

the two norms. In practice however, the corner rarely gives the optimal solution, so

it is best to use it as a rough guide. Constructing the L-curve for our problem, we

34

104

10 3

10 2

10 1

100

100

101

Residual norm || A x’in

x’out

||2

Sol

utio

n no

rm ||

x’ in

||2

Corner at λ = 6.7× 10 5

Increasing λ

Best fit λ = 1.6× 10 3

-

- - - -

Figure 3.4: L-curve for the Hilbert operator. The curve is a parametric log-log plotof the solution norm vs. the residual norm for different regularization parameter λ inthe direction shown. Most references suggest the optimal point is at the corner asshown in red, but physical problems often have the ‘best’ solution elsewhere.

find a value of λ = 6.7× 10−5 at the corner. The solution then is:

x′λ=6.7×10−5 =[

0.9298 1.5388 0.2284 0.8474 1.4818]T

(3.32)

x′λ=1.6×10−3 =[

0.9800 1.0661 1.0103 0.9773 0.9474]T

(3.33)

∼= xin. (3.34)

By looking at some more values near the corner, we find that the solution closer to

our ‘true’ solution actually has λ = 1.6× 10−3. So we see that we can in fact recover

sensible results even for badly conditioned problems.

3.3.1 An alternate interpretation

There is an alternative picture to justify the Tikhonov regularization scheme. Recog-

nizing that it is the small singular values that cause the difficulties, we can imagine

35

applying a filter on the singular values when we construct the inverse in eqn. (3.25).

Applying a Lorentzian filter to the reciprocal singular values, we get:

1

σi

→(

σ2i

σ2i + λ2

)σ−1

i (3.35)

For σi >> λ, the filter has little effect, whereas if σi << λ, then σ−1i → λ−1, limiting

the unstable spectrum. We see that the two views are equivalent since we can ana-

lytically solve the Tikhonov minimization (eqn. (3.31)). For a given λ, the function

is minimized if x′out is constructed using filtered coefficients of eqn. (3.35) instead of

the reciprocal singular values σ−1i . Different regularization schemes effectively change

how we evaluate the filtering coefficients. For example, if we take our Hilbert operator

and increase N to 100, we find the spectrum of singular values as shown in figure

3.5. Because of the distinctive corner in the spectrum, we might use an aggressive

strategy here and simply truncate beyond the 20th singular value. This is known as

the truncated singular value decomposition (TSVD) regularization scheme. (Note:

this scheme alone would not work for the Hilbert problem because the remaining σ’s

would still give a condition number of 1017.) Most physical inverse problems do not

have these obvious clusters, making hard truncation more difficult, so Tikhonov is re-

ally a good general strategy to use. In figure 3.6 we show the spectrum for the inverse

photonic problem (see chapter 6) using a Guassian output mode. This spectrum is

more representative of real world inverse problems.

3.4 Conclusion

In this chapter, we explored the reasons why there is a real fundamental distinction

between a forward problem and an inverse problem. In particular, the notion of

stability against random perturbations is what sets the two apart, and we gave the

condition number as a quantity that helps us identify ill-conditioning. For complete-

ness, we now elaborate on a subtle point that we commented on earlier in passing.

We motivated the need to find a fundamental distinction between forward and inverse

36

0 10 20 30 40 50 60 70 80 90 100

102

10 4

10 6

10 8

10 10

10 12

10 14

10 16

10 18

-

-

-

-

-

-

-

-

-

Figure 3.5: The spectrum of singular values for the 100× 100 Hilbert operator. Notethe distinct corner, showing an obvious point where we can perform a truncation.

problems due to the inherent symmetry of the two problems, i.e. one is the inverse of

the other. It should now be clear that neither the Fourier transform nor its inverse

can be considered an inverse problem, because they are both unitary, so C = 1. For

an inverse or ill-posed problem, C >> 1.

Of course, the condition number of the forward and inverse problem are the same

as we have defined it (since it is just the ratio of the largest to smallest singular

values), so the spectrum of singular values does not break the symmetry between the

two problems. So what actually breaks the symmetry? It turns out to be the special

status given to the random fluctuation or noise. The forward problem is defined

as the one that is stable against changes caused by random fluctuations. However,

given a large condition number, stability against noise necessarily implies it will be

‘unstable’ to a different form of perturbation. Of course, we usually do not use the

term ‘instability,’ but instead we use the term ‘sensitivity’ in this context. Historically,

this makes sense in how one studies physics. To model a physical system, we vary its

parameters and measure its effects. A model is good if it makes good predictions about

37

Figure 3.6: The spectrum of singular values for the inverse photonic problem usinga Gaussian shaped desired mode. In contrast to figure 3.5, we find no obvious placefor a hard truncation.

the effects. Given a new physical system we are trying to model, if any small noise

(i.e. a perturbation to the system the experimentalist cannot control) will create a

large disturbance in the effect we can measure, it will be very difficult to come up with

a model. What we need is a system that is sensitive to controllable and systematic

variations to the input, so the effects can be readily observed with adequate signal-to-

noise ratio. By its very nature, most problems studied are stable (in the sense given

here) against most forms of noise. Any physical model derived based on experimental

results will necessarily reflect this process. Therefore, we do expect ‘noise’ vectors to

have large projections onto the ‘bad’ singular vectors in physical problems, using our

linear algebra language.

3.4.1 Parameter estimation vs. design

We conclude this chapter by making an observation about the difficulties in transfer-

ring over from standard inverse problems to PBG device design. First, most standard

inverse problems can assume implicitly the existence of a solution even if the prob-

38

lem formally does not guarantee you a solution [21]. Going back to the cause and

effect picture, you are measuring a real effect from a cause that necessarily exists.

The problem as we saw is that noisy measurements hide the underlying cause. If

you really cannot reconstruct the cause from the measured effect, it is probably an

indication that the model is wrong. As applied to a design paradigm, that is not the

case. The ‘desired effect’ has not been observed or measured. It is a mere figment of

our imagination, so to speak. We will see that this is a much more serious problem for

design purposes. If we encounter a design problem where we encounter a non-existent

solution situation, we would like to make strong claims to that effect, but we cannot

do so because the Tikhonov regularization scheme is not really the most appropriate

for the PBG problem. This brings us to our second observation. The solution we

seek in the PBG inverse problem is the dielectric function, and their norms are not

necessarily small, particularly if there are discontinuities. Physically, η cannot take

on negative values, but Tikhonov would be happy accommodating negative values as

long as they are small. Fortunately, we can use the insight developed in this chapter

to implement a much more appropriate regularization scheme for the PBG problem.

We expand on these ideas and develop the necessary tools in the next chapter.

39

Chapter 4

Convex Optimization

At the conclusion of chapter 3, we suggested that a natural question to ask is whether

Tikhonov regularization is really the best choice for the purpose of the inverse pho-

tonic problem. We learned that regularization is a way to impose additional con-

straints on an under-determined (rank-deficient) system so that an ill-posed problem

becomes well-posed. Therefore, given an actual physical problem, one ought to be

able to customize a more appropriate set of constraints based on the relevant physics.

Suppose the desired solution has a fairly sizeable norm. Under Tikhonov, is there

a possibility that we might ‘regularize out’ valid solutions because the norm con-

straint is not sophisticated enough? For an even worse effect, given our application,

a solution with negative values would still have a small norm, but in the case of pho-

tonic materials, the solution (representing the dielectric function) would not even be

feasible. More succinctly, requiring a small norm does not correlate with valid and

desirable photonic device designs. A more suitable approach places upper and lower

bounds on the value of the dielectric function that correspond to air and semiconduc-

tor. We will derive in chapter 6 the inverse photonic problem, but for now, consider

the regularized problem given these constraints of the form:

minη|Aη − b|2

subject to ηmin ¹ F−1η ¹ ηmax

(4.1)

40

where F−1 is the inverse fourier transform operator. Equation (4.1) is a quadratic

minimization problem with linear inequality constraints, and falls under the general

category of convex optimization problems. Casting this problem in the form of a

convex optimization (CO) is powerful because rigorous bounds have been derived on

the optimality of their solutions [1] using interior point methods. Therefore, with the

only constraint being that the dielectric function take on physically realizable values,

we can prove rigorously whether any dielectric function exists that would support a

given target mode, as indicated by the magnitude of the residual norm. This chapter

is devoted to developing our implementation of the convex optimization algorithm.

4.1 Introduction

We have all encountered optimization problems, and our first exposure to them is

likely to have been in a calculus course. Given some function f(x) defined over some

interval x ∈ [a, b], find where the maximum (or minimum) value of f occurs along

that interval. In the language we will use below, we call x the ‘optimization variable,’

and f the ‘objective function.’ Restricting x on the interval [a, b] can be viewed as

an ‘inequality constraint’ on the optimization variable. We can locate local extrema

since they satisfy the following condition:

df

dx

∣∣∣∣xe

= 0 (4.2)

The sign of the second derivative evaluated at those points {xe} classifies the kind

of extremum (maximum, minimum, or inflection point) at those points. To find the

global optimum, we evaluate f at all the local extrema, plus the end points, and then

choose from among these the optimal value (x∗), and we call x∗ the solution to the

optimization problem1. For arbitrary functions, the problem becomes more difficult

as eqn. (4.2) may not have analytical solutions. Numerical optimization in 1D is

1Here we follow Boyd’s notation, and x∗ does not denote the complex conjugate of x. Boyd onlydeals with real-valued variables and functions, so the notation is fine. Later in this chapter whenwe deal with complex variables, we will use η to denote the complex conjugate of η.

41

relatively straightforward, as we can just plot the function and visually determine the

extremum points. However, the problem scales unfavorably as the dimensionality of

x increases.

minimize fo(x)

subject to fi(x) ≤ 0, i = 1, ..., m

gj(x) = 0, j = 1, ..., p

, (4.3)

where f0 : Rn → R is the objective function, x ∈ Rn is now an n-dimensional

optimization variable, and {fi : Rn → R} and {gi : Rn →R} define the m inequality

constraints and p equality constraints respectively that x must satisfy in order to be

a valid solution. We refer to any x that does not satisfy all the constraints as an

‘infeasible point.’ The problem of maximizing an objective function is achieved by

simply reversing its sign.

An optimization problem is called a ‘convex optimization’ problem if it satisfies

the extra requirement that f0 and {fi} are convex functions (which we will define

in the next section), and {gi} are affine functions. Furthermore, the set of feasible

points must also form a convex set. The special property for this class of problem is

that any local minimum is by definition also the global minimum, and thus solves the

optimization problem.

4.1.1 Organization

As with the simple one-dimensional variable optimization, we will find the first and

second derivatives play critical roles in these optimization algorithms, so we begin

with the definition of multidimensional derivatives in section 4.2. For our applica-

tion, we will need to extend the treatment to include the use of complex variables,

which have some minor complications we will address. We will formally define convex

functions in section 4.3 and discuss some of their properties, highlighting those that

help with understanding the optimization problem. With the mathematical tools suf-

ficiently developed, we can then explain how to implement the convex optimization

42

method in section 4.4. We explain how to select descent directions and the line search

routine for an unconstrained optimization first, and then show how constraints can

be incorporated. Again, our primary concern in the presentation here is not to be

mathematically rigorous, but rather make it accessible for engineers and physicists

looking for a softer entry point. With the exception of the modification required for

functions with complex variables, the material in this chapter is adapted from the

textbook by Boyd and Vandenberghe [1], where they provide a much more thorough

and rigorous treatment of convex optimization methods.

4.2 Derivatives

For a real-valued function f : Rn →R, the definition of the gradient is

∇f(x)i ≡ ∂f(x)

∂xi

, i = 1, . . . , n, (4.4)

provided that the partial derivatives evaluated at x exist, and where ∇f(x) is written

as a column vector. The first-order Taylor approximation of the function f near x is

f(y) ≈ f(x) +∇f(x)T (y − x) (4.5)

which is an affine function of y.

The second derivative of a real-valued function f : Rn → R is called a Hessian

matrix, denoted ∇2f(x), with the matrix elements given by:

∇2f(x)ij ≡ ∂2f(x)

∂xi∂xj

, i = 1, . . . , n, i = 1, . . . , n, (4.6)

provided that f is twice differentiable at x and the partial derivatives are evaluated

at x. The second-order Taylor approximation of the function f near x is

f(y) ≈ f(x) +∇f(x)T (y − x) +1

2(y − x)T∇2f(x)(y − x) (4.7)

43

We now derive some shorthand notation for taking derivatives of matrix equations

representing functions f : Rn →R.

For functions that are linear in x:

f(x) = aT x = xT a (4.8)

=∑

i

aixi (4.9)

∇f(x)i ≡ ∂f

∂xi

(4.10)

= ai (4.11)

∴ ∇f(x) = a. (4.12)

The Hessian is obviously 0 in this case.

For functions that are quadratic in x:

f(x) = xT Bx =∑i,j

xibijxj (4.13)

∇f(x)i ≡ ∂f

∂xi

(4.14)

= xT ∂Bx

∂xi

+∂xT

∂xi

Bx (4.15)

= xT [B]coli + eT

i

∑

k

bikxk (4.16)

=∑

k

(bki + bik)xk (4.17)

∇f(x) =(B + BT

)x, (4.18)

where [B]coli denotes the ith column of the matrix B and ei is ith basis-vector. For the

44

Hessian, we obtain

∇f(x)ij ≡ ∂2f

∂xi∂xj

(4.19)

=∂

∂xi

∇f(x)j (4.20)

=∂

∂xi

[(B + BT )x

]j

(4.21)

=∂

∂xi

∑i

(bji + bij)x (4.22)

= bji + bij (4.23)

∇2f(x) = B + BT . (4.24)

For composite functions of the form h(x) = g(f(x)) such that h : Rn → R, with

f : Rn →R, and g : R→ R, the chain rule for gradients is:

∇h(x) = g′(f(x))∇f(x). (4.25)

The chain rule for the Hessian is evaluated to:

∇2h(x) = g′(f(x))∇2f(x) + g′′(f(x))∇f(x)∇f(x)T . (4.26)

The chain rules will be very useful for incorporating the barrier functions to impose

inequality constraints.

4.2.1 Complex variables

For the photonic problem, the optimization variable η will in general be complex.

Therefore, our matrices and vectors will be complex as well. Our objective function

f(η) : Cn → R is actually the 2-norm of a complex residual |Aη− b|2. We have found

few resources that deal explicitly with complex derivatives. Petersen and Pedersen’s

reference [25] shows that we can treat η and η as independent variables, and then

the generalized complex gradient is found by taking the derivatives with respect to

45

η. This expression for the gradient is suitable for use with gradient descent methods.

∇f(η) ≡ 2∂f(η, η)

∂η(4.27)

For linear functions, we have

f(η) = 2|a†η| = a†η + η†a (4.28)

= a†η + ηT a (4.29)

∇f ≡ 2∂ηT a

∂η(4.30)

= 2a. (4.31)

For quadratic functions, we have

f(η) = η†A†Aη (4.32)

= ηT A†Aη (4.33)

∇f = 2A†Aη. (4.34)

The Hessian for the quadratic function is

∇2f(η) = 2A†A. (4.35)

Unfortunately, it turns out that we cannot define a chain rule for differentiating

complex variables. Part of the reason is that a real-valued function of a complex

variable is strictly not differentiable because it does not satisfy the Cauchy-Riemann

equations, which state for f(η) = f(x + iy) = u(x, y) + iv(x, y), where u, v are real

valued functions of the real variables x = <(η), y = =(η)

∂u(x, y)

∂x=

∂v(x, y)

∂y

∂u(x, y)

∂y= −∂v(x, y)

∂x

. (4.36)

46

A real valued function implies v = 0, so in order to satisfy Cauchy-Riemann u cannot

depend on x or y. The linear and quadratic functions happen to be special cases for

which these can be defined, but in general, we cannot evaluate complex gradients and

Hessians directly. However, we can treat <(η) and =(η) as independent real variables

so now we have a function f(x, y) : R2n → R. All of the previous and subsequent

results can now be applied, provided we define our functions in this form. We return

to our linear and quadratic matrix functions and see what the equivalent structure

looks like.

We define a Cn →R2n transformation for a complex column vector:

[η] → x

y

≡ [ξ]. (4.37)

The adjoint operation of the vector η → η† becomes ξ → ξT . This definition preserves

the norm of the vector.

η†η = ξT ξ (4.38)

(xT − iyT )(x + iy) = [xT yT ]

x

y

(4.39)

xT x + yT y = xT x + yT y (4.40)

∵ xT y = yT x (4.41)

For the function to be real-valued, vector-vector products must come in adjoint pairs,

i.e.,

η†1η2 + η†2η1 → [xT1 yT

1 ]

x2

y2

+ [xT

2 yT2 ]

x1

y1

(4.42)

i(η†1η2 − η†2η1) → [yT1 − xT

1 ]

x2

y2

− [yT

2 − xT2 ]

x1

y1

(4.43)

where we have used iη† = i(xT − iyT ) = yT − i(−xT ) in the second relation. We must

47

of course be cautious that if the function is not real-valued, this transformation breaks

down (e.g., a general dot product of 2 complex valued vectors will give a complex

number), so it is not accommodated here.

For matrix multiplications, we define the transformation A ∈ Cn×n → A ∈ R2n×2n

as follows:

[A] → Ar −Ai

Ai Ar

, (4.44)

where Ar = <(A), and Ai = =(A). As before, the Hermitian adjoint becomes the

simple transpose operation. Matrix-vector multiplications are preserved:

Aη = Aξ (4.45)

(Ar + iAi)(x + iy) =

Ar −Ai

Ai Ar

x

y

(4.46)

(Arx− Aiy) + i (Aix + Ary) =

Arx− Aiy

Aix + Ary

. (4.47)

In appendix B, we will show that using the chain rule for the log barrier function

with the generalized complex gradient definition is incompatible with our definition

here for a simple linear constraint function. For the remainder of the thesis, we will

not explicitly write out the conformal mapping from Cn to R2n.

4.3 Convex Sets and Functions

A set SC is convex if the line segment between any two members of the set x1 and

x2 also lies in SC . The geometric representation is shown in figure 4.2. Important

examples of convex sets are hyperplanes which have the form {x|aT x = b} and half-

spaces {x|aT x < b}, where a ∈ Rn, a 6= 0, and b ∈ R. Other common convex

sets include spheres, cones, and polyhedra. The set defined by the intersection of

two convex sets is also convex, so intersection preserves convexity. This is important

in our definition of a convex optimization problem, since as long as our constraint

48

-5 0 5-5

-4

-3

-2

-1

0

1

2

3

4

5

Figure 4.1: The chord connecting any two points of a convex function must lie abovethe function if the function is convex. The curve on the top is clearly convex. Thebottom curve is an upside down Gaussian. Even though it has a single local minimumthat is also the global minimum, the function is not convex as shown. The functionlies both above and below a connecting chord.

functions are convex, we are guaranteed a convex feasible set.

A function f(x) : Rn → R is convex if the domain of f is a convex set, and the

function satisfies the following relation:

f(αx + (1− α)y) ≤ αf(x) + (1− α)f(y) (4.48)

for all x, y ∈ Rn, and with the scalar α ≥ 0. For a given x, y pair, a parametric

plot of α on the right hand side of the inequality corresponds to the chord connecting

f(x) to f(y). We can provide a graphical interpretation of the convexity condition

as a function where for any pair of points the function lies below the chord joining

the pair of points, as shown in figure 4.1. A function is strictly convex if strict

inequality holds for all x 6= y in eqn. (4.48). By definition, a function f is concave if

49

(a) (b) (c)

Figure 4.2: Examples of convex and non-convex sets. The set of points in a polygon(a) and in an ellipse (b) are both convex. The star shape is not convex since theline connecting two points in the set passes through a region that is not in the set(highlighted in red).

-f is convex. For the rest of this chapter, we will only consider functions for which

the domain of f spans all of Rn, so the domain of f is always a convex set. Some

important examples of convex functions include

• Exponential eax is convex on R for any a ∈ R

• Logarithm log(x) is convex on 0 < x ∈ R

• Norms on Rn

• Linear, Affine, and Quadratic functions on Rn

• Non-negative weighted sums of convex functions f(x) = α1f1(x) + α2f2(x) for

f1, f2 convex functions and α1, α2 ≥ 0.

4.3.1 Convexity conditions

Suppose f is differentiable such that its gradient exists. The function f is convex if

and only if

f(y) ≥ f(x) +∇f(x)T (y − x) (4.49)

We recognize the right side of eqn. (4.49) as the multidimensional version of the lin-

ear Taylor series expansion of the function f about x. From this property of convex

functions, we observe two consequences. First, linearization of the function underesti-

mates it everywhere (see figure 4.3). This means the first-order Taylor approximation

50

Figure 4.3: Graphical representation of the linearization of a convex function.

is a global underestimator of f . In addition, when ∇f(x) = 0, f(y) ≥ f(x) for all y,

so x is the local and the global minimum of the function. The second-order condition

equivalent to f ′′(x) ≥ 0 for the 1D case is that ∇2f(x) º 0, i.e. the Hessian is positive

semi-definite.

4.4 Gradient and Newton Methods

We consider optimization problems for which we do not have an analytical solution,

and therefore must use a numerical (and iterative) algorithm to solve the problem.

We want an algorithm whose performance is independent of the starting condition,

and rapidly converges to the optimal solution. Starting with some initial non-optimal

51

point x(0), each iterate (k) of the algorithm gives us an x(k) such that f0(x(k)) → f0(x

∗)

as k → ∞, where x∗ is the ‘true’ solution that optimizes our objective function. In

practice, the iterations terminate once some specified tolerance level is reached, i.e.

f(x(k)) − f(x∗) < ε. In general, this would be difficult to estimate, but because of

convexity it allows us to evaluate bounds on how far from optimal the final solution

is. At each iteration, the intermediate solution is updated via the following general

relation:

x(k+1) = x(k) + t(k)∆x(k) (4.50)

where ∆x(k) is an unnormalized vector in Rn known as a ‘step direction’ and t(k) > 0

is a scalar called the ‘step size’ for the kth iteration. Different algorithms will have

different methods for determining the step directions and step sizes. We consider only

descent methods here for which f(x(k+1)) ≤ f(x(k)), with equality only if f(x(k)) is

optimized. The general algorithm can be described as follows:

• Initialize: Obtain a feasible starting point x

• Repeat

1. Determine the step direction ∆x.

2. Line search. Choose a step size t > 0.

3. Update x → x + t∆x.

4. Evaluate stopping criterion at the new x.

• until stopping criterion is satisfied.

4.4.1 Unconstrained optimization

To illustrate these ideas, we begin by considering an optimization without constraints.

The popular gradient descent method uses the negative of the gradient (−∇f) eval-

uated at the intermediate point x(k) as the step direction. The stopping criterion

is usually of the form |∇f |2 ≤ ξ, where ξ is small and positive. Using the negative

52

gradient guarantees that our step direction is a descent direction, and for simple prob-

lems it is easy to implement in practice. Unfortunately, for ill-conditioned problems,

this method does not converge in practice. However, since the gradient is easy to

visualize, we include it here to help illustrate the second step of the algorithm, the

line search.

Backtracking Line Search

The line search is used to choose how far to step in the descent direction (once that

is determined). In principle, one can do an exact line search of the following form:

mint>0

f(x(k) + t∆x) (4.51)

which is a 1D minimization problem. However, in practice, inexact methods are

used because they are easier to implement without suffering a loss in performance.

Inexact methods aim to simply reduce the function by some sufficient amount, and

the backtracking line search is the one we will use. The algorithm depends on two

parameters α, β, with 0 < α < 0.5 and 0 < β < 1. It is called backtracking because

it first assumes a full step size of t = 1, and if the step does not lead to some

sufficient decrease in the objective function, then the step size is decreased by a

factor β so that t → βt (see figure 4.4). The sufficient decrease condition can be

written mathematically as:

f(x + t∆x) < f(x) + αt∇f(x)T ∆x. (4.52)

For small enough t, this condition must be true because we only consider descent

directions. The parameter α sets the amount of decrease in the objective function we

will accept as a percentage of the linear prediction (which, due to convexity, provides

a lower bound). Practical backtracking algorithms tend to have α between 0.01 and

0.3, and β between 0.1 and 0.8, with a small value of β corresponding to a coarser

grained search of the minimum.

53

1.5 1 0.5 0 0.5 1

4

3

2

1

0

1

Number of Backtracking Steps:5

t=0t=t0

t=1

t=β

t=β2 t=β3 t=β4

t=1

4

{α∆f

∆f

New x

Figure 4.4: The linear approximation (red) to the objective function (black) expandedaround x = 0.64. The blue line shows the backtracking condition as accepting afraction α of the predicted decrease by linear extrapolation. After four backtrackingsteps, t = β4, and the value of the objective function has decreased enough.

As a simple example we choose the following 1D objective function:

f(x) = eγx + e−γx − 2, (4.53)

where γ is a parameter we will adjust to show the various behaviors of the gradient

descent method. The optimal value of x∗ is 0, and f(x∗) ≡ p∗ = 0. To illustrate the

idea of backtracking, we first show in figure 4.4 the objective function for γ = 1.25,

and choose as an initial point x0 = 0.64. The function is linearized at x0 in red, and

the blue line shows the backtracking condition for α = 0.2. A full step in the step

direction (t = 1) takes us to x = −1.5803, shown as a red star in the figure. As

illustrated, the region where the backtracking condition (eqn. (4.52)) is satisfied is

for t ∈ [0, t0]. Increasing α will decrease the size of the valid t region. The task for

the line search algorithm is to find a valid t. It backtracks from a starting value of

54

γ Number of Iterations Mean number of backtracking steps0.125 241 11.25 12 412.5 22 26

Table 4.1: Summary of gradient method performance using an objective function(eqn. 4.53) with 3 different sharpness parameter γ.

Figure 4.5: Convergence rate of different γ for gradient method.

1 until it enters the valid region. In this example, it backtracks 4 steps before the

function has ‘decreased enough,’ and the value of x for the next iterate is −0.2694.

If we go ahead and continue with the optimization, we find the solution converges to

x∗ = −1.87×10−6 in 12 iterations, and each iteration takes on average 4 backtracking

steps during the line search. If we now attenuate γ to 0.125 and repeat, we find the

solution converges to x∗ = 3.14×10−4 after 241 iterations, without having to backtrack

at any iteration. For γ = 12.5, it takes 22 iterations to find x∗ = 3.01×10−8, and each

iteration takes on average 26 backtracking steps, with a maximum of 52 at the first

iteration. These results are summarized in table 4.1, and the convergence rates are

depicted graphically for the three cases in figure 4.5. The problem with the gradient

method is that |∆x| = |∇f |. When γ is small, even a full step is too small to cause

substantial reduction in the objective function, while for γ too big, it leads to such

55

0 0.2 0.4 0.6 0.8

0

2

4

6

x 103

(b)

γ = 0.125

1 0 1

1

0

1

2

x 104 γ = 12.5

(a)

Figure 4.6: (a) For large γ, the norm of the gradient is too large, so a t = 1 step sizewould actually take us to x = −3.7 × 104 (well beyond the axis on the plot). Thisleads to a large number of backtracking steps in the line search. (b) On the otherhand, a small γ would give gradients with small norms, so small that a full t = 1step will still give only minimum improvement, necessitating many iterations beforeconvergence.

an enormous step that the backtracking must go through many iterations to return

to the valid t region (see figure 4.6). For ill-conditioned multi-dimensional problems,

we effectively have an enormous range of γ in different directions, so practically for

ill-conditioned problems the gradient method never converges.

Newton’s Method

A much better method that uses the second derivative information as well is New-

ton’s method, provided of course that the objective function is twice differentiable.

Newton’s method uses the Newton step as the step direction:

∆xnewton = −∇2f(x)−1∇f(x), (4.54)

56

0.5 0.6 0.7

1500

2000

2500

3000

3500

4000

4500

(a)

γ = 12.5

-0.5 0 0.5 1

0

0.5

1

1.5

(b)

γ = 1.25

-0.20

0.20.4

0.60.8

0

2

4

6

8

x 10

-3

(c)

γ = 0.125

Figure 4.7: (a) For γ = 12.5, the norm of the Newton step (marked by the magentaasterisk) is 0.08, compared to the |∇f | = O(104). (b) For γ = 1.25, the quadraticapproximation becomes quite good, and the full Newton step does satisfy the back-tracking exit criterion. (c) γ = 0.125, where the fit is even better, illustrating thequadratic convergence phase.

where ∇2f(x) is the Hessian matrix as defined previously in eqn. (4.6). The super-

script −1 denotes matrix inversion. The Newton step can be interpreted as the step

that minimizes the second order Taylor expansion of the objective function about the

point x(k) (see figure 4.7). For an unconstrained quadratic objective function then,

the Newton step exactly minimizes the objective function. The stopping criterion

using Newton’s method is the quadratic norm of the Newton step as defined by the

Hessian (also known as the Newton decrement),

λ(x) =(∆xT

newton∇2f(x)∆xnewton

)1/2(4.55)

Using Newton’s method, we repeat the same optimization of our objective func-

tion with the 3 different values of γ. The results are shown in figure 4.8 and table

4.2. Newton’s method benefits from the more rapid quadratic convergence, for as

we get closer and closer to the minimum, the accuracy of the second-order approxi-

mation improves. For this particular objective function, we see that we never have

57

Figure 4.8: (a) For γ = 12.5, we see the distinct corner at 9th iteration, showing thebeginning of the quadratic convergence phase. (b) For γ = 1.25, the final value off(x) is actually 0 to within machine precision, but shown as 10−15 for reference. (c)The same holds true for γ = 0.125.

γ Number of Iterations Mean number of backtracking steps0.125 3 01.25 4 012.5 11 0

Table 4.2: Summary of Newton method performance using an objective function (eqn.4.53) with 3 different sharpness parameters γ.

to backtrack, which is an indication that the Newton step includes some informa-

tion about the magnitude of the step size, as opposed to choosing a step direction

based on the gradient alone. Compared to the gradient method, we see a significant

increase in computational overhead (matrix formation and inversion), but given the

ill-conditioning of our problem, gradient methods simply do not converge. In the case

of the photonic design problem, the Hessian itself will sometimes be ill-conditioned

as well, but we can use a truncated SVD pseudoinverse to calculate the Newton step.

58

4.4.2 Incorporating constraints: barrier method

Consider the following constrained optimization problem in 1D:

minimize f(x) = eγx + e−γx − 2 (4.56)

subject to x0 − x < 0 , x0 = 0.75. (4.57)

We have constructed the problem so that the solution to the constrained problem

lies at the boundary at x = 0.75. In order to incorporate inequality constraints, the

objective function is modified to include barrier functions that impose a prohibitively

costly penalty for violating the constraints. The barrier function that we will use is

the logarithmic barrier function, and we make use of the fact that the log diverges

near 0, meaning:

limx→0+

log x → −∞. (4.58)

Recall the set of inequality constraints from our optimization problem (eqn. (4.3))

require fi(x) ≤ 0 for all i. The logarithmic barrier function is defined to be

φ(x) ≡ −m∑

i=1

log(−fi(x)) (4.59)

Using the chain rule (eqn. (4.25) and (4.26)), we can write down the expression for

the gradient and Hessian for the log barrier function.

∇φ(x) =m∑

i=1

1

−fi(x)∇fi(x), (4.60)

∇2φ(x) =m∑

i=1

1

fi(x)2∇fi(x)∇fi(x)T +

m∑i=1

1

−fi(x)∇2fi(x) (4.61)

If we modify our objective function such that we instead minimize

f0(x) +

(1

δ

) m∑i=1

− log(−fi(x)) (4.62)

59

-0.8 -0.6 -0.4 -0.2 0 0.2-5

0

5

10

15

20

25

δ = 0.2δ = 1δ = 5Ideal

Figure 4.9: The log barrier function for various δ’s. The black dotted line is the idealstep barrier. Notice the largest of these (δ = 5) gives the closest approximation tothe ideal function.

we see that a violation of the inequality constraints will not minimize the objective

function. Therefore, the minimum of this new problem will automatically satisfy the

inequality constraints. The parameter δ controls how steep the barrier is. We plot

the barrier function for various δ’s in figure 4.9. As δ → ∞ , the barrier function

has no effect on the objective function for x in the feasible set, so this modified

problem becomes exactly the original problem. For finite δ, the modified problem is

only an approximation, so the optimal point of eqn. (4.62) is not the optimal point

of eqn. (4.3). This is illustrated in figure 4.10. The difficulty with a large δ is that

the overall function becomes difficult to minimize even using Newton’s method. Boyd

attributes this to the rapidly varying Hessian for the logarithmic barrier function near

the constraint boundary. Therefore, unless you are close to the boundary (where the

solution likely lies), a second-order Taylor expansion is a poor fit to the modified

problem, leading to poor convergence. A smaller δ will increase the region where

the expansion is valid, but yields a solution that is less accurate. Figure 4.11 shows

the quadratic fit to our modified objective function with a moderate δ = 5000. Far

from the boundary for large δ, the barrier contribution is insignificant. Therefore,

the Newton step ignores the barrier and acts as though there were no constraints.

60

1 2 30

0.05

0.1

0.15

0.2

(a)

δ = 10

1 2 30

0.05

0.1

0.15

0.2

(b)

δ = 100

1 2 30

0.05

0.1

0.15

0.2

δ = 5000

(c)

Figure 4.10: The modified objective function (red) for various δ’s. The constraint isthat x ≥ 0.75, and the optimal point is at the boundary (shown as a solid verticalblack line near the left edge of the box). The original objective function (black) hasγ = 0.125. The barrier function is the dotted red line. (a) δ = 10 The modifiedproblem is a poor approximation of the original. The optimal x∗ is around 2. (b)δ = 100 Slight improvement of the approximation, with x∗ ≈ 1. (c) δ = 5000 gives amuch better approximation.

However, this would take us out of the feasible set (main figure in figure 4.11). We have

the same problem here as we did with the gradient method then, as we potentially

may backtrack many iterations to return to the feasible set. Closer to the boundary,

the quadratic fit becomes quite good, as the barrier has an appreciable effect on the

modified function. The inset of figure 4.11 shows in blue the second-order fit and

Newton steps in that region. Of course, for large δ that means we are already very

close to the boundary, i.e. the solution of the optimization problem. A priori, we

would have no way of knowing where that fast converging region is.

The problem is overcome by a process called centering, where a succession of these

modified problems are solved with δ increasing with each centering step (δi+1 = µδi).

We solve the modified optimization problem using some small initial δ0 by Newton’s

method. The solution of a centering step is used as the starting point of the next

centering step. This ensures we are close to the region where Newton’s method

converges rapidly, as long as we don’t increase δ too quickly. This is reflected in the

61

parameter µ. If µ is too large, then we will have less centering steps, but each step

will require more iterations before it converges. If µ is too small, then we will require

many centering steps. Typical implementations take µ between 10 and 20.

For our implementation of the convex optimizer, we use for our line search routine

α = 0.125, and β = 0.9. Our stopping criterion is ε = 10−20, and µ = 20. Our

optimization problems uses 15 centering steps and within each centering step, the

solution will usually converge after less than 10 Newton steps.

4.5 Conclusion

In this chapter, we summarized some of the important tools needed for numerical op-

timization of multidimensional problems. Using a simple 1D example, we visualized

how the gradient descent algorithm and Newton’s method minimize a given objective

function. The caveat is that our intuition may or may not extend into N dimensions.

We can certainly imagine fitting the objective function with a hyper-paraboloid sur-

face, and minimizing that as the Newton step. In higher dimensions, it just shows

that the gradient has more directions to be incorrect about, so in practical problems

it is of little use.

We have now outlined all the tools we need to solve our photonic regularization

problem. We do not have equality constraints in our example, but those can be

satisfied by solving a set of KKT equations, as described in detail in Boyd. A final

note is that with these interior point methods, it is important to first find a feasible

point (i.e. satisfies all m inequality constraints and n equality constraints) as the

first iterate. Our constraints are simple enough that we can always construct one by

inspection (take a uniform slab of dielectric with an average index of refraction). In

general, there are algorithms loosely based on these techniques that serve as feasible

point finders.

62

0 0.5 1 1.5

0

0.01

0.02

0.03

0.04

0.05

Figure 4.11: Using the δ = 5000 barrier function, we show in the main body thequadratic fit of the objective function. The green and blue lines show the second-order fit about x = 1.785 and x = 0.885. The green and blue asterisks show theexpansion point and the Newton step, outside of the feasible set. The inset showssimilar quadratic expansions in blue, illustrating a good fit where near the minimum.

63

Part II

Device Design

64

Chapter 5

Photonic Bandgap Devices: AnOverview

Having developed the computational tools necessary for the design problem, we

now return our focus to photonic bandgap (PBG) materials. Certainly, ever since

Yablonovitch’s publication [26] in 1987 the amount of research into PBG materials

has been extraordinary. Even at its inception, it was anticipated that PBG materials

would have a profound effect on semiconductor devices because of their ability to

inhibit spontaneous emission rates. One goal of this chapter is to provide a general

overview of some of the diverse applications that make use of these PBG materials as

a functional device. It is well beyond the scope of this chapter to do a comprehensive

review of the topic, so the devices described here will naturally be biased towards our

own interests and experiences, but the idea is to provide a sense of where our work

fits in within the field.

5.1 Introduction

The term photonic bandgap refers to the range of frequencies for which electromag-

netic waves are not allowed to propagate within the material, so incident radiation

on the surface will be perfectly reflected. We can locate these bandgaps by solv-

ing the Helmholtz equation (eqn. (2.15)) and looking at the allowed eigenvalues as

demonstrated in chapter 2. A complete bandgap means the frequency is forbidden for

all propagating directions. Much of the early work involved looking at these ‘bulk’

65

photonic crystal materials and verifying the existence of bandgaps [11, 27, 28, 29, 30]

in various lattice geometries. A lot of work was done searching for the geometry that

led to the largest bandgaps. Since waves cannot propagate through the material, they

act like perfect mirrors at the forbidden frequencies.

The idea of introducing intentional defects to produce localized states [31, 32] came

shortly thereafter, and the possibility of trapped states lead naturally to the idea of

optical cavities and waveguides. Of course, if we think of bulk photonic crystals as

perfect mirrors, then it seems reasonable to think you can ‘trap’ and ‘guide’ light

within slabs of these photonic crystals. PBG materials promised to ‘mold the flow

of light’ as suggested by the title of Joannopoulos’ book [6]. To date, a huge part

of the challenge has been in making these idealized devices. The ‘holy grail’ device

requires the use of complete bandgap materials, but a complete 3D bandgap with

controllable defects at optical frequencies is extremely difficult to fabricate, though

there have been recent demonstrations [33, 34]. The reason for the difficulty is that

the periodicity required for optical PBG materials is on the order of the optical

wavelength, so the features on these devices are in the nanometer scale. By far

the more common paradigm is to make quasi-2D PBG devices, where periodicity is

introduced only in 2D (xy plane), and localization in the out-of-plane (z) direction is

achieved not via the bandgap effect but by dielectric contrast (like conventional optical

waveguides). Most of these are fabricated by the use of electron beam lithography

and a combination of wet and dry etching techniques. Other fabrication techniques

include self-assembly type approaches, but they are not reviewed here. The ease of

fabrication of these planar photonic crystal devices is offset by the loss mechanism in

the third dimension. There are two main types of defect that we consider here: the

point defect and the line defect. We will see that these serve as building blocks for

many important devices, particularly in the area of integrated photonics.

66

Figure 5.1: Nominal Geometry of a W1 photonic crystal waveguide.

5.2 Building Blocks

The basic building blocks for a photonic circuit are the waveguides and resonators.

One of the problems with existing lightwave circuits is that conventional ridge waveg-

uides require a large radius of curvature to minimize bend losses, which limits how

compact these devices can be. Photonic crystal waveguides (PCWs) promise to over-

come this limitation, making ultra-compact optical devices possible. The most basic

conceptualization of the PCW is by removing a single line of holes, hence the name

line defect. Light within the defect region is surrounded by a PBG material so it

can only propagate along the defect region. We show such a waveguide structure for

a hexagonal lattice known as the W1 waveguide in figure 5.1. However, achieving

the benefits of this device in the planar configuration is still the subject of current

research. Some guidelines on how their behavior translates from an infinite height

approximation to a finite height slab structure are found here [35, 36]. Early work

on these planar PCWs varied different parameters in an attempt to get some basic

desirable properties, such as single mode operation [37, 38], or to control the frequen-

cies of the waveguide modes [39]. Controlling losses in these PCWs [40] and PCW

bends [41, 42] became an important consideration, as conventional planar lightwave

circuits outperform PCWs considerably.

Besides the guiding functionality of the device, one of the most interesting prop-

67

erties of the PCW is their dispersion characteristics. In particular, the group ve-

locity in these waveguides were found to be orders of magnitude less than ordinary

waveguides and even show anomalous behavior. Johnson et al. [43] proposed mak-

ing ultra compact and efficient modulators using PCWs within Mach-Zehnder based

devices by taking advantage of this property. Kuipers’ group have devised a beauti-

ful experimental setup [44] to measure and verify this PCW dispersion. There was

tremendous excitement about these PCWs, leading some to claim that they “can

design light paths with made-to-order dispersion”(emphasis added)[45], although all

that has been demonstrated is that dispersion is a function of the geometry. The

purpose of design is of course to advance from dependence to control. The distinction

is important because in a PCW, pulse distortion due to its dispersion can occur over

much shorter length scales as compared with conventional waveguides, so the unique

feature of the PCW becomes another component that needs to be managed unless

one can control the dispersion. Dispersion management in general is a very important

issue in the field of telecommunications, where pulse broadening limits the bit rate

of a communication channel (since adjacent 1’s and 0’s broaden and blend together

making them indistinguishable). Some efforts at controlling the dispersion properties

can be found in these references [46, 47, 48], but to our knowledge, there have been no

demonstrations of arbitrary dispersion design. For integrated optics applications, the

effectiveness of add/drop filters and other such components would be severely com-

promised without managing pulse distortion on the chip. The problem we address

using our technique is to design a PCW that achieves an arbitrary target dispersion

relation.

The other building block we will examine is the optical resonator, or the PBG

cavity. This is conceptualized in a manner similar to the PCW, where we have some

region surrounded by PBG materials acting as perfect reflectors. Light within that

region becomes trapped. PBG cavities can be point defects (a break in the lattice

symmetry at a single lattice point), or enclose a larger region, such as with ring

resonators by arranging PCWs in a suitable configuration. Optical resonators in

general are useful for use as filters, and the figure of merit is the quality factor Q,

68

which is a measure of the linewidth of the transmission spectrum of the cavity. Point

defect PBG cavities can have high Qs (∼ O(106) or higher) with mode volume of the

order of the wavelength of light. Such cavities are desirable for many applications,

such as low-threshold lasers [49], or as add/drop filters [50, 51] when integrated with

PCWs for de/multiplexing applications, or for chemical detection in lab-on-a-chip

type applications [52, 53]. High-Q small-mode volume cavities are also crucial for

achieving strong coupling between an atom and the optical field in cavity quantum

electrodynamics (cQED) experiments. Our group’s initial venture into the world of

PBG came as a collaboration with Axel Scherer’s group, studying the feasibility of

using PBG cavities in the strong coupling regime [54].

5.3 PBG and Atomic Physics

Cavity quantum electrodynamics (cQED) provides a setting in which atoms interact

with the electromagnetic field within an optical resonator. It is one of the first ex-

perimentally realizable systems whereby one can quantitatively study the dynamics

of an open quantum system under continuous observation, and as such provides a

means for testing the laws of quantum measurement, such as quantum trajectory

theory [55]. In particular, experimental advances in recent years have crossed the

threshold where the intrinsic quantum mechanical atom-field interaction dominates

the dissipative and decoherence mechanisms into what is known as the strong cou-

pling regime [56]. Under strong coupling, the presence of a single atom or photon

is sufficient to affect the properties of the system, enabling the possibility of single

atom switching and single photon nonlinear optics. Other exotic applications within

strong coupling include quantum state mapping between atomic and optical states

[57], which is critical for the realization of quantum information processing [58].

An emerging reality in the laboratory is the use of nanofabricated optical res-

onators such as PBG cavities [59], microdisks [60], and microtoroids [61] for cQED

experiments with cold atoms. By incorporating a network of photonic crystal devices

with single trapped atoms [58] operated under the strong coupling regime [62], scal-

69

able quantum information processing can be performed in a quantum network. To

motivate the shift to these nanofabricated resonators, consider the pioneering works

in strong coupling that make use of high finesse Fabry-Perot cavities. These cavities

are much more sensitive to vibrational noise that is not common-mode relative to the

two mirrors. To maintain the proper coupling with the atom of interest, the distance

between the two mirrors of the Fabry-Perot must be stabilized to ∼10−15 m via active

servo control because of the narrow linewidth. The desire to operate at the single pho-

ton sensitivity makes it impossible to derive an error signal directly from the ‘physics’

signal for the servo lock. Therefore, an additional auxiliary frequency stabilized laser

is required, and its wavelength must be far enough detuned so that it does not inter-

act with the atom in the cavity. The operating complexity continues to increase as

frequency stabilizing this auxiliary laser involves another optical cavity, which also

requires another servo control. This extensive amount of labor overhead that precedes

each experiment renders this paradigm unscalable for networking purposes.

5.3.1 Waveguide dispersion design

In a quantum networking scenario, individual quantum nodes (physically realized

using coupled atom-cavity systems) are connected via photonic crystal waveguides.

Quantum information stored in an atom can be mapped onto a photon using cQED,

and the information transmitted to the next node as the photon performs another

cQED interaction with the trapped atom there [57]. The successful implementation of

the system will require attention to both the losses as well as the dispersion character-

istics in the waveguide. The desire to limit losses in a PCW as it goes through bends

and other optical elements is certainly not unique to this application, as we reviewed

earlier. Perhaps more unique to the application here is a demanding constraint on

the dispersion characteristics of the waveguide, because the photon emission from

one quantum node needs to be temporally (and spatially) mode matched for proper

excitation of the next node. Losses and pulse distortion will compromise the coupling

efficiency to the atom and cause a decrease in transmission fidelity. What is desired

70

is the ability to arbitrarily control the dispersion of a waveguide right on the atom

chip to compensate for the pulse shape distortion. In contrast to the usual notion

of dispersion compensation, where it is only the slope within some small k-vector

window that gets adjusted, we seek to specify the full dispersion curve. We have not

seen in the literature any method that enables arbitrary dispersion engineering of a

PCW.

5.3.2 Large defect region cavity design

The second element to the scheme involves the optical cavity. In coupling a single

atom to a point defect nanocavity, much of the emphasis has been on finding high-Q

and small-mode volume cavities in order to achieve strong coupling [63, 64]. However,

what is also desired is a large air (vacuum) opening in which the atom can be trapped,

since we wish to minimize the interaction between the atom and the PBG material.

Other applications where a large air hole defect is desirable include lab-on-a-chip

devices, where we wish to maximize the amount of analyte that can be introduced

into the optical excitation region for analyte detection [52] and on-chip optical spec-

troscopy [53]. In these applications, the important property to consider is that the

atom (or chemical) interacts with the electric field that is localized in the air (or

vacuum) region. Consider again the h1 structure where the electric field is localized

near the defect region. Given a particular bulk hole radius (rbulk), if we attempt to

increase the defect region by increasing the radius of the defect hole (rdefect), then as

we transition from rdefect < rbulk to rdefect > rbulk, we change from an acceptor type

mode (figure 5.2a,b) to a donor type mode (figure 5.2c,d) [31].

However, for the applications we have contemplated, a donor mode would not be

appropriate since an electric field node is located at the center of the evacuated region.

One strategy would be to increase rbulk along with rdefect, but that would compromise

the physical integrity of the device (see figure 5.3), as well as increase the scattering

losses [54]. What we seek then is a re-distribution of the layers surrounding the defect

hole in a way that increases the overall amount of dielectric (to an acceptable level of

71

-3 -2 -1 0 1 2 3

2

0

-2

1

-1

(b)

-3 -2 -1 0 1 2 3

2

0

-2

1

-1

(c)-3 -2 -1 0 1 2 3

2

0

-2

1

-1

(d)-3 -2 -1 0 1 2 3

2

0

-2

1

-1

(a)

Figure 5.2: (a) Electric Field intensity of an acceptor mode. Note the field maximumat the center of the air defect region. (b) Magnetic Field intensity. (c) Electric Fieldintensity of a donor mode. Note the field minimum at the center of the air defectregion. (d) Magnetic Field intensity.

Figure 5.3: h1 geometry with large bulk hole radius. Fabrication uncertainty couldlead to entire sections of photonic crystal collapsing.

72

mechanical strength), while still retaining the desirable qualities of an acceptor mode.

Quantitatively, we can require

ηD ≡∫D η(r)dr

AD>

∫B η(r)dr

AB≡ ηB (5.1)

where A is the area, and D and B designate the defect and bulk region respectively.

Other groups have demonstrated that we do not require a localized defect within a

perfectly periodic lattice in order to localize a mode [65, 66], although it is not clear

whether it is possible to violate the simple donor/acceptor mode intuition by rear-

ranging the bulk region. Even if it were possible, intuition fails to provide guidance

as to how the redistribution should happen. We examine this problem using our ap-

proach.

73

Chapter 6

Inverting the Helmholtz Equation

6.1 Introduction

In this chapter, we formalize our idea for using an inverse problem based approach

to photonic device design. We begin in section 6.2 by surveying the field of PBG

design using inverse problem based approaches in the literature, comparing the various

methods with the approach we have adopted. This will frame the results presented

in these final chapters within the greater context of this emerging field of research.

In section 6.3, we derive the inverse Helmholtz equation we need to solve in order to

address the design problems motivated in section 5.3. We provide a simple proof-of-

principle example in section 6.4 to show that the inverse equations are in fact correct,

and also the importance of regularization for solving this problem. We conclude

this chapter with a first look at what happens when we ask for a mode that is not

supported by a physically realizable dielectric function. This problem will be explored

more extensively in chapter 7.

6.2 Inverse Problem Based Design

As we reviewed in chapter 5, there are many applications envisioned for PBG devices.

However, the traditional design paradigm is really based on trial and error, which is

not as suitable for applications. In the existing paradigm, different geometries are pro-

posed and studied (numerically or experimentally), and the effects catalogued. As the

74

field progresses, the understanding of the fundamental scientific principles increases,

and we begin to collect together useful effects that can be adapted for useful appli-

cations. Devices are improved by repeatedly varying different parameters to existing

designs. After much trial and error, more intuition regarding the design problem is

developed, so that, with a lot of human ingenuity, one hopes that subsequent ‘trials’

will less frequently turn out to be ‘errors.’ The other problem is that despite the

many excellent devices that have been developed using this approach, one can usually

not be certain how much room for improvement exists for the device. In other words,

we are uncertain how optimally it is designed given the current manufacturing/cost

constraints. Even if we somehow know that we are not optimal, the traditional design

approach also does not give us a systematic or algorithmic way that lead us to the

better/best designs.

An inverse problem based method is the exact reverse of the traditional method.

Rather than systematically varying the causes (dielectric function) to catalogue the

resulting effects (optical properties), we purposefully choose the desired effect and

look for the unknown cause. As explained in chapter 3, the starting point is the

effect, rather than the cause. We will review some methods that claim to be inverse

problem based, but are effectively automated trial and error methods.

In an applications driven design paradigm, we imagine designing a device that will

optimally perform some desired function. The starting point of the design process is

focused on the end application, rather than the best known solution to date. This

paradigm shift thus lends itself readily to the inverse problem formulation. Given

some desired performance criterion suitable for the application, one needs to first

develop a performance metric P(Hi(r), ωi(r), η(r)) that may be a function of multiple

eigenmodes and eigenfrequencies, and even the structure itself. The goal of optimal

PBG device design is to find a structure that maximizes this function. It should

be clear from our use of the term optimal in chapter 4 that we mean the global

rather than the local optimum in this context, though we will see that this is often

not how the term is applied in the literature. We can generally assume that this Pmetric is obtainable in that under any design approach, it is required for evaluating

75

the performance of various designs. A design method that produces these optimal

devices is considered an optimal design method (and to be clear, in principle this

does not necessarily need to be an inverse problem based approach). An optimal

inverse problem based design method combines both of these features. The underlying

assumption is that if we can somehow find the field that optimizes the objective

function (which can be a difficult problem in itself), an inverse problem method will

allow us to solve for the dielectric that produces the desired properties. This was the

idea behind the work (in the 2D approximation) by Geremia et al. [67], and is in fact

the predecessor to the work in this thesis. It turns out there were some fundamental

errors with the way the inverse problem was formulated, and these are explored and

discussed in appendix D.

A brief survey of the field

Despite increased interest and efforts at developing this inverse problem based design

paradigm in recent years, there are still only a few papers that have been published.

For a more comprehensive review of the inverse problem approach to PBG design,

the reader is referred to an article by Burger et al. [68], but the emphasis there is

on the topology optimization approach, particularly those using the level set method.

The journal Inverse Problems has many electromagnetic type inverse problems, but

few directly applicable to photonic crystals or PBG devices in particular. An early

treatment by Popov et al. [69] in this journal of a 1D photonic crystal (i.e. alternating

layers of dielectric) scattering problem shows that given the reflection coefficient,

the index of refraction can be determined assuming ‘practically sensible conditions’,

which is of course a regularization condition. The 2D scattering problem is treated

by Ammari et al. [70], and both of these are quite heavy on the mathematics, and are

more concerned with parameter estimation than design. The interest there is much

more heavily biased towards the mathematics of the inverse problem rather than the

physics, and this is typical of the papers one finds in that journal to date. The

first direct application of inverse problem methods to the PBG community was by

76

Dobson and Cox, whose work focused on finding unit cell geometries that maximized

the bandgap in the TM [71] and TE [72] polarizations. Their technique uses a

gradient-based algorithm, and they found that their final ‘optimal’ designs were quite

sensitive to the targeted mid-frequency of the bandgap (which of course indicates that

they have only reached local optima). The method in the TM polarization required

a structure with an existing bandgap, but the later work removed this restriction.

6.2.1 Genetic algorithms

Sanchez-Dehesa’s group has recently published some results based on an inverse prob-

lem method for designing high efficiency waveguide couplers [73] and waveguide de-

multiplexers [74]. The demultiplexer design for an incoming signal with wavelengths

1.5 µm and 1.55 µm gives > 45dB crosstalk suppression and > 75% coupling effi-

ciency. However, the method they use is a genetic algorithm, where the geometry is

parameterized and the only variable is whether the cylindrical rods of fixed size and

location remain or are removed. Of course, using a genetic algorithm (GA) does not

solve an inverse problem as we have defined the term in this thesis. It is systematic

trial and error, with the errors generally discarded. The other comment is that GA’s

generally do give good results for globally optimizing arbitrary functions with many

local maxima and minima, as long as the design space is kept small. Therefore, this

method cannot guarantee the optimality of the design over all possible structures

since the design space must be heavily parameterized.

The technique by Gheorma et al. expands on their approach by allowing ‘aperiodic’

structures, so the location of these cylindrical rods are not fixed [66]. They found

that a straight-forward application of the GA did not converge well given the extra

degrees of freedom, so they used an adaptive algorithm, but it is of course still not

an inverse problem method as we define it. The key idea here that is consistent with

our work is that they give up the notion of an overall lattice periodicity to achieve

improved performance. This is a bit of a break from the traditional view of PBG

devices, where the bandgap effect plays a vital role in the design. Our results on the

77

enlarged defect region cavity design have similar philosophical underpinnings. The

other relevant result here is that they allude to the idea of not actually obtaining

the mode they had designed for, but attribute this effect to numerical (i.e. series

truncation) errors and other constraints rather than as a fundamental limitation. To

our knowledge, this is the only other paper that makes this observation explicitly,

although the importance of this property was not fully appreciated.

6.2.2 Topology optimization methods

Topology optimization methods have also been used for minimizing losses in a waveg-

uide bend [75], T-junction [76], and to improve the directional transmission properties

of a waveguide termination [77]. In contrast to the GA methods, the dielectric func-

tion is not parameterized, but discretized into finite elements, where each element

can take on any value. Therefore any design within the discretization bandwidth is

within the design space. This aspect is similar to the domain of our design space.

The optimization routine uses the iterative Method of Moving Asymptotes (MMA)

optimizer which is a local gradient based algorithm, and requires the determination

of the sensitivities of the objective function to the design variables at each iteration.

However, computing the various sensitivities directly is not feasible, so linear approxi-

mations to the model are used. Linearization is necessary even though the Helmholtz

equation is linear in the dielectric function because the objective function is not lin-

ear in each of the discretized elements. Based on the perturbation theory of linear

operators [78], we expect this function can be highly non-linear. The validity and

consequences of the approximations made are not discussed, but given the accuracy

issues with just solving the forward problem, more caution should be exercised here.

In addition, there are some known problems associated with the method. It has

been observed that the design algorithm stops converging as the resolution of the grid

is increased [79]. This effect is attributed to the numerical instability of calculating the

gradient as the ill-conditioning increases. This is consistent with what we discussed in

chapter 4 regarding gradient descent methods for ill-conditioned problems. Finally,

78

since this is a gradient based method, it is also limited to finding locally optimal

designs.

6.2.3 Level set method

Closely related to topology optimization is the level set method [80]. The level set

function is a way to define an interface between distinct media, so it is a more ideal

way of describing the dielectric function. Briefly, the zero crossings of the level set

function define the boundaries between ε1 and ε2. The optimization is done by iter-

atively updating the level set function by solving a Hamilton-Jacobi equation where

the velocity term is chosen to climb the gradient to the objective function. Similar

approximation issues found with the topology optimization methods are encountered

here. This technique was recently applied to maximizing the bandgap for a 2D square

lattice with great success for the TM polarization, and to a lesser degree for the TE

polarization [81].

6.2.4 Analytical inversion of waveguide modes

Of the various papers in the literature, the work by Englund et al. [64] is probably the

most similar to ours in spirit, and received positive review from the community [82].

They designed high-Q small-mode volume cavities by analytical solution of the inverse

problem. By restricting the set of target modes to an expansion of the waveguide

modes with a slowly varying envelope, they reduce the inverse problem to a 1D

problem and analytically solve for the dielectric function along the waveguiding axis.

The off-axis structure along the line defect region is reconstructed from the solution

of the dielectric function along the line by assuming cylindrical defects. They solved

for both a Gaussian and a sinc function envelope modulated defect mode with Q’s of

1.6×106 and 4.3×106, and mode volume (in (λn)3) of 0.85 and 1.43 respectively. The

results while promising have some limitations that were not addressed. Particularly

for the sinc function design, the output mode actually did not resemble the target

mode. In fact, comparing their figures 6(c) and 6(f), the sinc function mode looks

79

more like the Gaussian mode. This is actually consistent with our finding, as we

believe this general inability to reach an arbitrary target is a ubiquitous problem.

6.2.5 Our approach

The first distinctive feature is that we derive the inverse problem from first principles,

i.e. ab initio. We make no additional approximations or linearizations beyond a

reduction of the problem from 3D to 2D. In contrast to the topology optimization

methods, we solve the full inverse problem exactly, rather than restricting ourselves to

local improvements to existing designs using approximate methods. The philosophy

behind the approach is that if you can specify what you want, our method will tell you

how to get it. Approximate methods limit the set of functionalities you can specify,

because they must be of the correct form.

The other distinctive feature of our approach is that we do not parameterize the

dielectric function at all in our design space. Any geometry commensurate with our

chosen bandwidth is within the design space. Using the convex optimization regu-

larization tool, we remove non-physically realizable values of the dielectric only. We

can include additional fabrication constraints without increasing the computational

domain if we so desire. Thus, we are not limited as in the GA approaches or the

waveguide expansion approach. On this point, it is more akin to the topology op-

timization method. A criticism of our approach might be that our designs are not

binary valued (as with level set methods), so they are not compatible with current

fabrication technology. As discussed in appendix C, any discretized dielectric function

should really be interpreted via its underlying continuous function. In many cases,

we would argue that these should not actually be considered binary valued anyway.

Secondly, even without this limitation, this is somewhat intentional. Consider a de-

vice requiring continuous valued dielectric functions that can improve performance

by several orders of magnitude. Existence of such a design would still be impor-

tant information to have. This may provide the right incentive to push fabrication

technology improvements in that direction. Of course graded index fibers are a sim-

80

ple example of such a dielectric function, so they are not fundamentally impossible

designs. Conversely, if existing designs already yield close to optimal performance,

then it is not worth spending more effort into looking for improvements. Either way,

there is value to our formulation. In that sense, it was our goal to construct a design

method that simply gives you the best device possible without particular regard for

current fabrication limitations, because those can improve with time and ingenuity.

It is possible to derive new insight into the problem, which is another contribution

of this method to the field. Furthermore, some forms of fabrication constraints can

still be modeled into the design problem as well, and we demonstrate that with our

enlarged defect cavity problem. We now derive the inverse equations for the design

problem.

6.3 Inverse Helmholtz Equation

Using the plane wave basis, the derivation of the inverse Helmholtz equation uses

the same mathematical principles that we used in obtaining the matrix form of the

forward problem (explicitly derived in section 2.2 and applied to defects using the

supercell method in section 2.3). Our restriction to TE polarized modes within a 2D

analysis described in chapter 2 is still in effect here. We make use of the completeness

and orthogonality of the plane waves here as well, but rather than solving for the hk

coefficients given an η(r), we solve for the ηk coefficients given some desired or target

field distribution Hm(r) instead.

6.3.1 Point defects

Working out the derivation for the point defect design, we expand the field in the

Fourier basis:

Hm(r) =∑κ

a(m)κ eiκ · rz (6.1)

81

Starting from the Helmholtz equation again, we left multiply with a plane wave and

integrate. ∫e−iγ · r

{ω2

m

c2Hm(r) = ∇× (η(r)∇×Hm(r))

}d2r

ω2m

c2a

(m)γ =

∫e−iγ · r

∇×

∑

k

ηkeik · r∇×∑κ

a(m)κ eiκ · r

d2r

ω2m

c2a

(m)γ =

∑

k, κ

a(m)κ ηk κ · (k + κ)

∫ei(k + κ − γ) · rd2r

≡ b(m)γ

Choosing to collapse the delta function with κ = γ − k

b(m)γ =

∑

k

[a

(m)

γ − kγ · (γ − k)

]ηk (6.2)

≡∑

k

A(m)

γkηk (6.3)

In anticipation of using Tikhonov regularization for solving the inverse problem, we

split up the dielectric function into two parts: an ‘initial’ geometry and a small

corrective part:

∑

k

A(m)

γkδηk =

ω2m

c2a

(m)γ −

∑

k′A

(m)γk′ η

(0)

k′ (6.4)

∑

k

A(m)

γkδηk ≡ β

(m)γ (6.5)

We use β and b to distinguish between the two cases. Just as we did with the

Helmholtz operator Θ(η) in equation (2.24), we have explicitly included a superscript

m on A, a, b, and β to emphasize their dependence on the desired mode Hm(r).

The importance of A’s dependence on the desired mode will be explored in section

7.2. The procedure for solving the inverse problem given some desired mode is to first

82

express it in the Fourier basis, and then forming the A matrix using eqn. (6.2) and

(6.3), and finally solve either eqn. (6.3) or (6.5) depending on which regularization

scheme we choose to utilize.

6.3.2 Line defects

For the waveguide dispersion, recall that there is a separate eigenvalue problem for

each wavevector qi along the propagation direction of interest. For each qi, we must

perform the same steps as above on the waveguide form of the Helmholtz equation

(eqn. (2.29)) to obtain the following inverse problem:

b(qi,m)

γ =∑

k

[a

(qi,m)

γ − kγ + qi · (γ − k + qi)

]ηk (6.6)

A(qi,m)

k, k′ ≡ a(qi,m)

k − k′(k − k′ + qi) · (k + qi) (6.7)

The solution ηk will have to simultaneously satisfy all Nq of these inverse equations,

where Nq is the number of wavevectors we will include in the dispersion curve. This

can be formally expressed by a vertical concatenation of the A(q) matrices and b(q)

vectors.

A ≡

A(q1)

A(q2)

...

A(qn)

, b ≡

b(q1)

b(q2)

...

b(qn)

, and β ≡

β(q1)

β(q2)

...

β(qn)

(6.8)

For the remainder of this chapter, unless required for clarity, we will omit many of

the cumbersome subscripts and superscripts on A, b, and β with the understanding

that, with a truncated basis, they can be treated like matrices and vectors.

83

6.4 Proof of Principle

In this section, we will work through a contrived problem as a proof of principle

demonstration, but also demonstrate the steps one would take in performing such

a design procedure. We first define our computational domain. We will choose a

hexagonal lattice of cylindrical air holes1 of radius r = 0.3a in dielectric using a

7a × 7a supercell, and include 19 reciprocal lattice vectors before truncating the

Fourier series. The total number of plane waves is 931 in this example, and a is the

lattice constant. As our illustration, we will ‘design’ a point defect cavity geometry

where the central air hole is refilled with a dielectric material. This is referred to

as the h1 defect. Using the set of 931 plane waves, we would normally construct an

Hm(r) with some suitably desired properties. In this case, we obtain the ‘desired

mode’ Hm(r) by explicitly solving the forward problem.

For this simple geometry, we can use the analytical expression for ηk. We follow

the method found in [3] to evaluate the transform. The reference gives the defect-free

coefficients:

ηG = ηdδ(|G|) + (ηa − ηd)2πr2

√3

(2J1(|G|r)|G|r

)

where ηd = 111.56

is the reciprocal dielectric constant of the dielectric (value is typical

of a semiconductor like AlGaAs), ηa = 1 represents air, and J1 is the Bessel function.

Evaluating only when k = G gives us the bulk symmetry. Adding the defect means we

need the expansion for δη(r) corresponding to filling in the air hole. A straightforward

modification of the above gives the required coefficients.

ηk = (ηd − ηa)2πr2

N1 ×N2

√3

(2J1(|k|r)|k|r

)

This is now evaluated for all k’s, and the factor N1 × N2 in the denominator is

due to integrating over the size of the entire supercell. We show in figure 6.1 the

underlying dielectric function. Using these coefficients, we construct the Helmholtz

1Recall the discussion in section 2.4 about the convergence issues of the plane wave method. Thuswe are actually considering the underlying continuous function of the nominal geometry. Refer toappendix C for additional details.

84

Figure 6.1: Underlying continuous dielectric function of the nominal h1 defect geom-etry.

operator using eqn. (2.28) and solve the eigenvalue problem. The localized mode is

located at the 50th band. Figure 6.2 shows the magnetic field of the localized mode.

We now take this H(r) and using only information about this field, attempt to get

back the original dielectric function. We form the inversion matrix A following eqn.

(6.2). Of course, after the discussion in chapter 3, it should not be surprising that

this problem is ill-posed. Nevertheless, just to illustrate, we can try to perform a QR

factorization to invert the A matrix and solve for ηk. The result is shown in figure 6.3,

and as we expect, it looks like noise that has been amplified, bearing no resemblance

to the actual dielectric function that produced this mode.

6.4.1 Tikhonov solution

We now use Tikhonov regularization to solve the inverse problem. As a demonstra-

tion, we imagine that we are only given this localized mode, and assume we have no

knowledge of what photonic crystals or bandgaps are at all. Solving the regularized

problem gives a solution shown in figure 6.4. The structure is a marked improvement

over the QR solution, and definitely suggests creating a periodic lattice of air holes

85

Figure 6.2: Real part of the magnetic field intensity for localized defect mode for theh1 defect geometry.

with a central defect, although there are still some areas near the edges that do not

quite resemble the exact solution. This can be partly attributed to the fact that the

solution norm is sizeable |ηk| = 0.37. To improve on the result, we now invoke the

perturbative form of the inverse equation (eqn. (6.5), and use the defect-free bulk

lattice as an initial geometry. This finally gives us the same geometry that we had

started out with, demonstrating that we have indeed solved the inverse problem. This

solution is shown in figure 6.5.

6.5 Simulating Design Errors

In the previous section, the entire scenario is, of course, rather artificial, since we

knew (by explicit construction) that some geometry must exist that will produce the

target mode. In an actual design problem, we would not be certain of that a priori.

We model this uncertainty by adding a small noise term to the target (h1) field. From

the discussion in chapter 3, we know that we are not guaranteed the existence of a

solution in an inverse problem, which means that some desired modes just simply

86

Figure 6.3: Solution to h1 inverse problem using a QR factorization to invert A.Notice the values of the dielectric function far exceed the original function.

cannot be supported. In this example, even if the perturbed field is not supported,

we would still like to recover the h1 geometry, because we know that the h1 geometry

reproduces the target field minus the small noise term. In the following example, the

noise corresponds to a 1% perturbation.

Using the perturbed field as the target field, we proceed to form the inverse prob-

lem as before with a bulk lattice starting point and solve using Tikhonov regulariza-

tion again. Using the same regularization parameter of λ = 3×10−3 as before yielded

a noisy solution similar to the QR factorized solution. Clearly, much more regulariza-

tion is required in this case. In figure 6.6, we show the solution δη(r) using λ = 3.73.

The residual norm in this case was 0.3143. We show both the real and imaginary

parts of δη(r), since there are some fairly significant contributions to the imaginary

part of the dielectric. Looking at the real part of δη(r), we find the dominant feature

is as we expected, which is to make the central air hole more dielectric-like. Notice

also in the dielectric function some fluctuations in the area surrounding the defect,

which we see is an attempt to accommodate the added noise term in the target field.

Of course, our model assumes that the dielectric is real-valued, so the imaginary parts

87

Figure 6.4: Solution to h1 inverse problem using Tikhonov regularization with noassumptions about the dielectric geometry. Solution suggests creating a periodiclattice of air holes with a central defect.

are particularly problematic. Again, because we knew a priori the ‘correct’ solution,

we can safely disregard the ripples in the real part and focus only on refilling the

central hole.

A general limitation of the Tikhonov scheme is that the solution will need to

be ‘interpreted’ to look for the most reasonable or feasible solution. In this case, if

we did not know what the correct geometry should have been, we would first have

dropped the imaginary parts, since we cannot do anything about those anyway, and

then started the next iteration by filling in the central hole (since it is the most

prominent feature). We would then redo the forward problem with the central hole

refilled, and compare with our target mode. If we were still not satisfied, we could

solve for the inverse problem again, but using the solution from the last iterate as

the ‘initial geometry’. This is the strategy we will use for the PCW dispersion design

problem. We would still not be guaranteed the solution, and we may well find the

desired field is not supported. To rigorously confirm this, we make use of the convex

optimization scheme developed in chapter 4.

88

(a) (b)

Figure 6.5: Solution to h1 inverse problem using Tikhonov regularization with aninitial defect-free lattice geometry. (a) Actual solution δη(r) to the inverse equation.(b) Full reconstructed solution η(r) = η0(r) + δη(r), which look identical to figure6.1. The regularization parameter used was λ = 3× 10−3, and the residual norm wasbelow 10−9.

6.5.1 Convex optimization regularization (COR)

The problem we need to solve, first shown without explanation in eqn. (4.1) becomes

the following:

minη|Aη − b|2

subject to ηmin ¹ F−1η ¹ ηmax

(6.9)

where F−1 is the inverse fourier transform operator. The symbol¹means component-

wise less than or equal to, so each discretized value of the real-spaced dielectric

function2 must lie between ηmin and ηmax. We explicitly set the imaginary part of

the dielectric function to zero by enforcing η∗k = η−k and using the transformation

in section 4.2.1. Notice that we are using b instead of β in the objective function,

which means we assume no knowledge of the defect-free lattice. We use the convex

optimization algorithm as described in chapter 4 to solve the constrained minimization

2The dielectric function used to generate this forward problem does not use the analytical ex-pression for ηk because truncation leads to overshoot. As shown in figure 6.1, ηmax > 1, which isstrictly unfeasible. Therefore, the actual values of η can exceed the bounds here. Refer to section8.2 for how this is handled.

89

Figure 6.6: Solution to the noisy h1 inverse problem using Tikhonov regularization.(a) Real part of δη(r). (b) Imaginary part of δη(r). The regularization parameterused was λ = 3.73, and the residual norm was below 0.314.

problem. The solution is shown in figure 6.7 with a residual norm of 0.2489. There

is a slight discrepancy between our solution and the original geometry within the

region where the target mode has very little intensity. Intuitively, this is sensible

if we think of the inverse problem in terms of its signal to noise ratio. Where the

mode has little to no intensity, there is insufficient signal to overcome the added

noise to reconstruct the desired dielectric completely. Without the added noise to

the target field, the COR reconstructs the dielectric function perfectly, as with the

Tikhonov regularization scheme with a residual norm at the machine precision level.

The difference is that we did not require prior knowledge of the defect-free lattice

using COR. We defer a more thorough comparison of the two schemes until section

7.4.1. With the added noise, both schemes gave reasonably close approximations to

the original geometry. The interpretation we ought to make here is that the noisy

mode is not supported by any physically realizable geometry. We can claim this

rigorously because the globally minimized residual norm to the solution of eqn. (6.9)

is non-zero. Therefore, no other geometry exists that can reduce the norm further (or

exactly solves the inverse problem). This means that we cannot track the noise that

has been introduced, and we will explore the role of the residual norm more closely

in the next chapter.

90

Figure 6.7: Solution to the noisy h1 inverse problem using CO regularization.

91

Chapter 7

From Inverse Problems to DeviceDesign

7.1 3:1 Waveguide Splitter

Encouraged by our analysis in the last section, we now attempt our first simple but

non-trivial design problem where we wish to have an incoming waveguide split into

2 branches with a ratio of 3 to 1, and then recombined. We retain the 7a × 7a

supercell geometry and construct a mode that complements the chosen hexagonal

lattice. The mode is created by first adopting a transverse Gaussian profile along

the splitter path, then modulating each segment with an appropriate plane wave.

To produce the 3:1 split, we attenuate the upper branch amplitude to 25% of the

unbranched intensity, and the lower branch to 75%, as in figure 7.1. We choose the

desired frequency (ωa/2πc) to be 0.2081, which lies in the middle of the bandgap.

We choose the simpler Tikhonov scheme and use the defect-free lattice as a starting

point again. Figure 7.2 shows the regularized solution with a residual norm of 3.4491

using a regularization parameter of 32.4. In figure 7.3 we show the L-curve for this

problem, where λcorner ≈ 1.5. The much more subtle ‘corner’ is more typical of a real

design problem, and we found (as we mentioned in section 3.3) that the best solution

is not near the corner anyway. The solution at λ = 1.5 is shown in figure 7.4, and

again, notice the scale on the colorbar and the substantial amount of noise in the

solution.

92

Figure 7.1: Target mode for the 3:1 splitter device.

In general, we try several parameters until a sensible solution is found. From our

experience, we often have to regularize beyond the L-curve corner, and ‘interpret’ the

result by smoothing out the noisy components. Generally, we look at a particular

solution and identify dominant features, and then propagate the smoothed solution

through the forward problem again. In this case (ignoring the imaginary parts again),

we find the real part of the dielectric suggests a geometry where the upper branch

has a hole radius that is half that of the bulk hole radius, and the lower branch

has completely filled holes as in figure 7.5. Propagating this geometry through

the forward problem yields the 3:1 split mode as shown in figure 7.6. Notice that

the actual mode we obtained was not identical to the original target mode, with the

obvious difference between the two being that the target mode was more localized

than the obtained mode. However, since our design goal was only to construct a

3:1 split waveguide, we can stop here. The discrepancy between the two should not

be surprising in light of the magnitude of the residual norm, as we discuss in the

following section.

93

Figure 7.2: Regularized solution of the 3:1 splitter inverse problem using a regular-ization parameter of 32.4

7.2 Residual Norm

Recall that the expression for our inverse equation comes from the Helmholtz equa-

tion:

∇×(η(r)∇×Hm(r)

)=

ω2

c2Hm(r) (7.1)

∴ Θ(η)h(m) =ω2

c2h(m) ≡ b (7.2)

Ah(m)

η = b and, (7.3)

Ah(m)

η = Θ(η)h(m) (7.4)

η(r) and Hm(r) are a special pairing since η(r) is the dielectric that supports Hm(r) as

an eigenmode. The equality in equation (7.1) only holds if the two are an ‘eigenpair’.

For some given solution ηsol we find in solving the inverse problem for which Aηsol 6= b,

it necessarily means that Hm(r) is not an eigenmode of ηsol. This finite residual norm

is significant for the design problem, because it means we are guaranteed not to

obtain our desired mode. The obvious question to ask is what exactly are we getting

(using a residual norm metric) when we don’t find an eigenpair? We can examine the

94

10 -1 100 10110 -0.9

10 -0.8

10 -0.7

10 -0.6

10 -0.5

10 -0.4

10 -0.3

10 -0.2

10 -0.1

100

Residual Norm

Sol

utio

n N

orm

L-curve for 3:1 Splitter Inverse Problem

Solution Presented

Approximate location of L-curve corner

Figure 7.3: L-curve for the 3:1 splitter showing the location and shape of the corneras well as the optimal solution.

left hand side of the equation using ηsol to form the Helmholtz operator Θ.

Ah(m)

ηsol − b 6= 0 (7.5)

Θ(ηsol)h(m) − b 6= 0 (7.6)

Since the forward problem is well-posed, we can always find the spectrum of eigen-

modes for any given dielectric geometry. Assume the following spectral decomposi-

95

Figure 7.4: 3:1 splitter δη(r) solution using regularization parameter from the L-curvecorner. The information in this ‘solution’ is practically useless.

tion:

Θ(ηsol)hηsoli =

ω(η)2

i

c2hηsol

i

Let h(m) ≡∑

i

αihηsoli

Then Θ(ηsol)h(m) − b = Θ(ηsol)h(m) − ω2m

c2h(m)

= Θ(ηsol)∑

i

αihηsoli − ω2

m

c2

∑i

αihηsoli

=∑

i

(ω2i − ω2

m)

c2αih

ηsoli

(7.7)

Even though minimizing the residual norm over (ηsol) may be the ‘best’ general strat-

egy for solving an ill-conditioned linear system of equations, it becomes clear that it

is not the most appropriate strategy for the purpose of the design problem. Using the

eigenvalue decomposition shows explicitly that the entire spectrum of eigenmodes of

ηsol contribute to the residual norm, whereas we only care that our desired mode be

close to a single eigenmode of ηsol. Furthermore, each eigenmode in the spectrum is

component-wise weighted by the ω2i term. Since the frequency of the modes of inter-

est usually lie within the first bandgap (i.e. relatively low frequencies), the residual

96

Figure 7.5: Actual η(r) used based on the output solution of the inverse problem (seefigure 7.2.

norm metric overemphasizes high frequency mode contributions. Therefore, the min-

imization has a bias towards dielectric geometries whose high frequency eigenmodes

are orthogonal to the desired mode at the expense of a stronger overlap of a single

eigenmode.

As an illustration, we can repeat our perturbed inverse problem using the noisy h1

mode as our target mode (only we add an even smaller perturbation (|n| = 10−3×|hm|)to the mode and renormalize). Rather than looking to solve the equation as we did

in section 6.5, we enter our exact solution ηh1k into |Aη− b| and evaluate the residual

norm. Even though it would produce our target mode minus the small bit of noise,

the residual is 1.59. Because of the high frequency components, we see that the

best solution to the physical problem does not even come close to solving the linear

equation.

In situations where the desired mode happens to be an exact eigenmode of some

geometry, the residual norm metric is fine, because it will simply find the comple-

mentary solution (as in our contrived example). The inverse equations find a solution

whose spectral decomposition of the target mode is a pure eigenmode, so it can afford

to ignore the high frequency mode spectrum. However, this is a rare occurrence, and

we will elaborate further in section 7.3. As we deviate further from an exact eigen-

97

Figure 7.6: Magnetic field distribution of the 3:1 splitter mode of interest supportedby the design of Fig. 7.5.

mode, the effect of the errors become more observable, as we saw in section 6.5. In

any realistic design problem then, what benefit can we get out of this approach? This

is precisely the phenomenon we see with the 3:1 waveguide splitter. We found that

the obtained eigenmode was actually quite different from the desired one, and yet it

achieved our goal. By examining the residual norm more closely, we now understand

that the optimality of our result depends somewhat fortuitously on the target mode’s

proximity to a feasible eigenmode. The more overlap between the target mode and a

feasible eigenmode, the less it needs to compromise with the high frequency modes.

However, we also see that despite this non-optimal bias, it still manages to find a

‘reasonable’ result. Particularly in situations where we truly have little intuition into

the problem, this will at least give us a good starting point. Therefore, we cannot

claim the resulting mode is optimal as we might have hoped, but we find that it is

still ‘better’ than what we may otherwise have.

7.3 Inverse Problem Based Design Flow

We began chapter 6 by motivating an inverse problem based design approach to obtain

‘optimal’ PBG structures, but so far we have really only outlined a single step: that

of solving the inverse problem. In the previous section, we showed how even this one

98

step is often not optimal, and we discovered the importance of a ‘good’ target mode.

In this section, we expand on the entire design process, highlighting other difficulties

to achieving optimality.

Recall that the idea of optimality is connected to a performance metric that is

some function of the desired field mode, and possibly of the dielectric function as

well. To find an optimal design implies finding the optimum of the performance

function, but there are unfortunately no guarantees that one can actually find the

global optimum of such a function. High-dimensional global optimization for arbitrary

functions is notoriously difficult (NP-hard). Therefore, the success of even the first

step (i.e. coming up with the target mode) will depend on the form of the function.

One can sometimes get around this by choosing to define the performance function in

a way that mimics the desired behavior, but will also have some nice features such as

convexity (and thus solvable by numerical optimization methods). One such example

can be found in correlating Fourier components within the light cone with the loss in

cavity Q factor [64, 9]. This restricts the types of performance criteria over which we

can optimize.

A second problem is that there may not be a unique optimum to our function. In

our 3:1 waveguide splitter example, we arbitrarily chose a parameter for the width

of the transverse Gaussian profile because it really did not matter. From a design

perspective, as long as the split was 3:1, we are somewhat ambivalent about what the

actual width needs to be, as long as the mode remains confined. As such, there would

be many such field distributions that are ‘optimum’ for the design goal. However,

the inverse problem approach demands the full specification of a single field. Which

one of these should we choose? If we can always find a dielectric that produces our

desired eigenmode, then this is not a problem. In that case, it simply means that

there are multiple designs that are all suitable. Unfortunately, if this is not the case

(and so far, it appears that this is the norm), then we have the problem of non-zero

residual norm. The question becomes which, if any, of the other field distributions

with different width profiles might have been valid, and thus form a solvable inverse

problem. There is no way of knowing unless we try them all, which starts to look like

99

trial and error and thus defeat the purpose of the approach.

7.3.1 Valid eigenmode landscape

The critical question to ask becomes how prevalent are these valid eigenmodes? Are

we more likely to come up with valid modes or invalid ones that do not readily lead to

a solution? If most modes one can design are in fact valid, or at least approximately

valid, then that is not likely to be a problem. Unfortunately, we can make some

strong arguments that the invalid regions are much more prevalent.

Consider a linear operator L such that Lx = λx. For small perturbations ∆L,

we know that the perturbation to the corresponding eigenvalues and eigenvectors are

bounded [78]. Consider the set of all perturbations having some norm |∆L| ≤ ξ,

and let the neighborhood of points (corresponding to normalized vectors) on the unit

hypersphere that bounds the rotation of a given eigenvector be denoted S. If we treat

this purely as a mathematical inverse eigenvalue problem, then we can access any new

eigenvector in S while bounding only the norm of ∆L. Any vector in the neighborhood

of an existing eigenvector is a valid eigenvector of some perturbed operator. Indeed, it

may seem strange (having framed our design problem in the linear algebra language)

to hear about vectors in CN that are unfeasible eigenvectors. However, we do not

have a purely mathematical inverse problem.

Consider again the Helmholtz operator

Θηk,k′ = ηk−k′k · k′. (7.8)

In contrast to our general operator L, the Helmholtz operator has N × N elements

and is parameterized by a vector of length N , so we have fewer degrees of freedom

to accommodate arbitrary changes to eigenvectors. The structure of the operator

means that only a few of these ∆L-type perturbations can take on the valid form of

∆Θ = δηk−k′k · k′, and even fewer of these have an expansion of η(r) that can take on

physically realizable values. Now, it is certainly true that there exist perturbations

to η(r) that keep you within S. In fact, the observation of robustness of devices to

100

Figure 7.7: Perturbation introduced to η(r).

fabrication uncertainty [65] is a good example. However, existence of these modes

does not reveal the density of these modes. Our claim is that, due to the rigid form of

the Helmholtz operator, the landscape of H(r) consists mostly of invalid eigenmodes.

Therefore, the density of valid modes in S is small. While we cannot prove this

rigorously, we have performed numerical simulations to test our theory.

First, we apply a small perturbation to our canonical h1 geometry by nomi-

nally increasing the radius of the central defect from 0 to 0.03. This corresponds

to |∆ηk/ηk| = 8.6 × 10−4 (see figure 7.7). We then solve the forward problem and

find the resulting perturbation to the defect mode |∆hk| = 0.013 (see figure 7.8).

Using the original h1 eigenvector, we now add to it a perturbation of a much smaller

magnitude (|∆ak| = 10−3), and proceed to form the inverse problem using this per-

turbed mode. We minimize the residual norm using the convex optimization scheme,

so a non-zero residual norm indicates definitively that the target mode is not a valid

eigenmode. In figure 7.9, we plot the distribution of the residual norm for 10000 of

these tests. None of these perturbed target ‘eigenvectors’ corresponded to physically

realizable operators. Therefore, the neighborhood surrounding a valid eigenmode ap-

pear to be mostly invalid eigenmodes, while the valid modes occupy a set of lower

dimensional hypersurfaces within the domain of H(r)’s. As we increase the resolution

of the computational grid, the dimensionality increases, further reducing the effective

101

Figure 7.8: Change in eigenmode as a result of perturbation to η(r).

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.350

100

200

300

400

500

600

700

800

Figure 7.9: Distribution of residual norms of perturbed inverse problems.

density of valid eigenmodes. If we look at figure 7.9 closely, we find the mean value of

the residual norm is around 1.15, whereas in section 6.5, the residual norm was closer

to 0.25. The number of plane waves used to generate this figure was 3721, whereas

we had less than 1000 before, which is another indication that this problem scales

badly with the dimension of the state space.

This dramatically changes our concept of the first part of our design process, i.e.

the performance optimization of a given mode. The performance metric does not

have a continuous domain (set of all normalized vectors, i.e. a hypersphere in CN),

but instead only has little pockets of validity. There is no way of knowing where these

102

pockets are a priori, and yet the optimization problem must take this into account.

Even when an exact eigenmode is ‘close’ to our optimized target mode, this distance

is not evaluated over the relevant performance metric. It is a general 2-norm distance

rather than a weighted distance function that is meaningful to the design goal. We

do not know how much of the desired property of the field is lost when we are forced

into the valid eigenmode. The limitation imposed by this ‘holey’ landscape prevents

the kind of precision implied by the performance optimization. Given our analysis, it

makes the optimization somewhat moot as we are not likely to benefit from it anyway.

We also believe this limitation is quite general despite our bandwidth limitation

here. Although computationally impractical, there is no fundamental limitation to

the resolution one can use in principle. The argument presented here still holds, and

we believe the regions of invalidity dominate more as the dimensionality increases. At

infinite resolution, we recover the exact Helmholtz equations, and thus this limitation

is clearly independent of the computational model. What we have observed poses a

formidable challenge to arbitrary and/or optimal design of PBG structures using an

inverse problem method. What is apparent is that this is not a turnkey type design

methodology, despite our stated goal of an algorithmic approach to PBG design.

7.4 Conclusion

One obvious improvement that deserves investigation is an alternative to the residual

norm as a minimization metric, as discussed throughout this chapter. Since we are

dealing with ill-posed problems, we must accept the fact that there is no existence

theorem. However, unlike ‘standard’ inverse problems (where one typically tries to do

parameter estimation of the system based on noisy measurements), lack of existence

is much more detrimental for our application, whereas lack of uniqueness is not. It

does not trouble us that multiple designs all give the same result, whereas the oil

prospector cares greatly where the rigs should be located, so a non-unique solution

to their inverse problem is much more problematic. When the target mode we seek is

not an exact eigenmode, we would like a technique to find a dielectric that supports

103

an eigenmode that is closest to the target mode. As it stands, the entire spectrum

contributes to the residual norm, and in a way that is non-optimal. Unfortunately, we

have not found a more suitable metric that we can efficiently compute at this point.

Until a better metric can be found that decouples the target from the rest of the spec-

trum, we have elucidated a fundamental limitation to arbitrary and optimal design

of photonic structures using an inverse problem based method. Our insight although

illustrated through a specific and limited model, turns out to be quite general. Our

analysis reveals that the underlying physics fundamentally forbid certain modes from

being physically realizable, regardless of human ingenuity. The problem of finding an

optimal structure for an arbitrary application, even confirming its optimality, remains

an outstanding question in the field of photonic design. Clearly, it is critical to check

the validity and quantify the performance of designs obtained through inverse prob-

lem methods. We have also established that this general approach is not a turn-key

method, but is potentially a very useful tool in instances where one has absolutely

no intuition as to how to meet a particular design goal. While the results may not

be optimal, they can spawn new geometries that serve as a starting point for other

design methods for further fine tuning.

7.4.1 Comparing Tikhonov and Convex Optimization Regu-

larization

Given the inadequacy of the residual norm, the advantage of the COR is not fully

realized. We had hoped that non-existence of the solution could be overcome by

finding the ‘next best’ solution that does exist. Currently, all COR does for you is

tell you that the designed mode is not realizable based on the size of the residual

norm. The solution it ends up giving you is not necessarily better or worse than

the Tikhonov scheme. As we saw with the noisy h1 problem, the original solution

had a relatively large residual norm, and represents a better solution than some

with smaller residual norms. The increased computational efforts make COR less

attractive in some circumstances. However, COR is able to handle more sophisticated

104

constraints, whereas Tikhonov cannot, so sometimes we may not have a choice. In

addition, COR is less subjective than Tikhonov. Part of the Tikhonov procedure

is looking at solutions at various regularization parameters and determining which

one is ‘best.’ We already showed that the L-curve is an objective but unreliable

method for finding the best regularization parameter. So that part of the scheme can

seem fairly subjective. In addition, the Tikhonov solution will return non-physically

realizable values, so the output will need to be fixed. There is some ‘slop’ inherent

in the process, which we may frown upon somewhat, although the residual norm

limitations show us that the entire design process is necessarily less rigorous than we

had hoped, so we should not be as concerned about it. The COR on the other hand

can be fully automated, because the output is guaranteed to be physically realizable.

This is especially important with iterative schemes that require many iterations. Both

methods will yield good but non-optimal designs, depending on how applicable the

residual norm is. The better a metric the residual norm is, the greater the advantage

with the COR. This is the reason why we will take finer steps with the COR iterative

scheme than with the Tikhonov iterative scheme in dealing with our cQED design

problems.

105

Chapter 8

Results

8.1 Waveguide Dispersion

We begin with the W1 waveguide, using a 9a×1a supercell geometry with a bulk hole

radius of 0.3a, and select 11 points along the Γ −M propagation direction (labeled

q) and truncate after 19 Brillouin zones. This gives us 171 plane waves per q-point

for a total of 1881 plane waves. We formulate the Helmholtz operator as in eqn.

(2.30) and solve to obtain the dispersion curve in the Γ−M direction. We choose the

curve with the anomalous dispersion characteristic, and as the primary objective1 we

seek to flatten the entire curve by 50% (see figure 8.1). As an additional objective

in anticipation of mode matching with a high-Q small-mode volume cavity, we also

require that the waveguide mode be more localized. The inverse problem is exactly

as set up in chapter 3, and shown in eqn. (6.8). We once again use the defect-free

lattice as η0(r), and since we do not have any additional constraints, we can use the

simpler Tikhonov scheme. The required dispersion relation ωi(qi) is embedded into

the set of βi’s. The target eigenmodes were constructed out of the initial eigenmodes

by compressing H(r) in the y-direction by 50% and padding the edges with zeros.

This mode was then smoothed out by using the Fourier coefficients (numerically

integrated) up to the truncation bandwidth and then renormalized. The resultant

1There is still no preliminary data on the specific form of the envelope function of a photonemission from an atom strongly coupled to a PBG-cavity, so at this point we do not know the preciseform of the dispersion curve we will need. As our demonstration of arbitrary design, we chose toflatten the anomalous curve because both anomalous dispersion and the flattening of dispersioncurves have greater relevance in the PCW community.

106

Figure 8.1: Dispersion relation for the W1 waveguide. The black x is the dispersioncurve of the nominal W1 waveguide, while the red triangles show the desired curve.

target field is now simply the truncated fourier series using those coefficients.

8.2 Enlarged Defect Cavity

The cavity problem has the added complication of the fabrication constraint (eqn.

(5.1)). This design problem cannot be solved using the Tikhonov regularization

scheme, since a small norm does not exclude the undesired designs. Our investment in

a more sophisticated regularization scheme allows us to incorporate these additional

design constraints as part of the regularization procedure (provided that they are in

the required convex form). We select a 5a×5a supercell geometry with a resolution of

12 points per lattice constant for a total of 3721 plane waves used. A smaller supercell

was chosen because we did not want the algorithm to simply decrease the holes at

the outer layers. We anticipate the algorithm may do that because the e-m field is

negligible there (hence making the supercell approximation valid). The smaller ge-

ometry also allows us to perform calculations at a higher resolution so we can resolve

107

Figure 8.2: 5× 5 h1 point defect cavity starting geometry.

finer details in the inverted geometry. In particular, rather than strictly using the

analytical expression for the Fourier coefficients, we use the FFT coefficients obtained

for the discontinuous dielectric function and broaden it by convolution with a sharp

Gaussian. Figure 8.3 shows the cross-section of the initial underlying dielectric func-

tion used. By defining the dielectric this way, we avoid Gibb’s phenomenon with the

Fourier series truncation, which is particularly important because the regularization

scheme we have chosen bounds the maximum and minimum values of the dielectric

to physically realizable values. Since we are testing a violation of the donor/acceptor

mode rule, we also want to get accurate values of ηD and ηB. We solve for the mode

of a structure with nominal bulk hole size of rbulk = 0.4a and rdefect = 0.3a (see figure

8.2)2, and use that localized mode as the target mode.

In contrast to the waveguide dispersion problem, here we only care about the

properties of the field, and not so much about the frequency. We can simply scale

the ‘lattice constant’ of the structure to accommodate a real-world frequency that

we would wish to use in a lab. We can therefore allow the frequency of the mode

2The actual ‘hole radii’ of the continuous function becomes smaller than these nominal valuesbecause of the convolution with the Gaussian. At this resolution, this was the largest nominalradius configuration we could accommodate without significant overshoot of the underlying dielectricfunction.

108

Figure 8.3: Cross-section view of dielectric function. The nominal structure and theactual structure is compared. Convolution with the Gaussian (pink curve) removesthe overshoot due to truncation, but bandwidth limitations prevent an analysis of thenominal structure. Otherwise, the design goal and constraints are analogous.

to be another optimization variable, with the constraint that the frequency remain

within the bandgap. This is another advantage to this scheme, as we no longer

need to have a correct target frequency (as in [67]) in order to obtain the correct

solution. To accommodate for the fabrication constraints and the variable frequency,

109

the regularized problem of eqn. (6.9) has to be modified slightly:

minη

∣∣∣Aη∣∣∣2

subject to ηmin ¹ F−1η ¹ ηmax

ηrD ≥ ηD

ηrB < ηD

ω2min

c2≤ ω2

c2≤ ω2

max

c2

where A = [ A | − b ] ,

η =

[ηT |ω

2

c2

]T

,

ηrD ≡

1

AD

∑DF−1η , and

ηrB ≡

1

AB

∑BF−1η

(8.1)

where the superscript T denotes the transpose. We choose ηmax = 1 and ηmin =

0.0796, andω2

min

c2= 1 and ω2

max

c2= 2.45.

8.2.1 The direct solution

When we attempt to solve the two problems, we find as expected large residual norms,

meaning that our target modes (and dispersion curve) are not supported. We can

solve the forward problem using the solution to the inverse problem as our design.

This gives a dispersion relation shown in figure 8.4, and a cavity mode that has only

72.9% overlap with the original target mode.

On the one hand, the results are quite encouraging, and in particular, the cavity

mode is surprisingly good given our discussion about the acceptor/donor mode regime.

On the other hand, the performance is not ‘optimal’ in the sense that we have not

reached our target perfectly in either case. The question will always remain whether

we have done the best we can do with a particular design, and that, as discussed in

chapter 7, is difficult to overcome. However, we now show an approach that can be

110

Figure 8.4: Initial result of the waveguide design. The green circles show the obtaineddispersion curve, while the red triangles show the target curve. The black x show theoriginal dispersion curve.

applied to these two problems that does yield even better results.

8.3 Iterative Approach

The method we will use to overcome these limitations involves an iterative approach.

While we still cannot make any optimality claims, it does achieve the specified objec-

tive to an astonishingly high degree. We know that the root problem is the sparsity

of the valid eigenmode landscape as discussed in section 7.3.1. The idea with the

iterative approach is to use known valid eigenmodes as starting points of the inverse

problem. We elaborate on the details as applied to each problem below.

111

8.3.1 Dispersion design

Given that our desired specifications are not physically realizable, it is time to make

some design compromises. For the waveguide dispersion design problem, we relax the

requirement that the eigenmodes be exactly as specified, and focus on the dispersion

relation requirement (i.e. the eigenvalues). After the initial step, we solve the forward

problem for the actual supported waveguide mode, and use that as the new target

mode of our next inverse problem iterate. The target eigenvalues, however, remain

as originally specified at each iteration. The iterative procedure proceeds as follows:

1. Use target modes and frequencies to formulate A and b.

2. Solve the inverse problem. Pick a regularization parameter that gives a reason-

able looking geometry.

3. Solve the forward problem using the interpreted result (as in section 6.5) from

the solution of the inverse problem.

4. Look for the eigenmodes of interest. Check the frequencies of the waveguide

modes. Continue if not satisfactory.

5. Use the new eigenmodes as the new target modes, but keep the original target

frequencies.

6. Repeat until successive designs no longer improve.

Here, the motivation for the iterative approach is in obtaining a new starting field to

define the updated inverse problem. This approach is inspired by the fact that we

then start with an eigenmode that is known to be valid, even though we intend to

perturb the eigenvalues. We know that the eigenmodes will be altered (if the inversion

is successful), so that the target eigenmode-eigenvalue pair will likely be invalid still.

However, without better insight, the prior valid eigenmode seems as good as any

other. Using our strategy, at each iteration, the inverse problem tries to balance the

112

Figure 8.5: Final result of the waveguide design. The blue circles show the finaldispersion curve, while the red triangles show the target curve. The black x show theoriginal dispersion curve. The inset shows the waveguide mode still well localized.

demand to change frequencies with the need to accommodate the current eigenmodes.

We let the result of the inverse problem tell us what the fields need to look like for the

next iteration, since we have no way of knowing what they should look like anyway.

The inverse problem ‘machinery’ itself will dictate how the dielectric needs to alter

in order to shift the eigenvalues in the desired manner, and we will simply track the

evolution of the eigenmode as it evolves. Each iteration brings us closer to the desired

frequencies, and we accommodate the evolution of the fields by updating the target

mode at each iteration.

The final waveguide design achieves a dispersion curve that compressed the origi-

nal curve by 57% instead of the target of 50% (see figure 8.5). The field mode remains

well localized, although we had not put any explicit restrictions on its form after the

first iterate. This is not entirely surprising, since we know in general that small shifts

in frequencies (eigenvalues) can be generated through small changes in the dielectric

(as demanded by the Tikhonov regularization), which then bound the changes in the

113

Figure 8.6: Final waveguide design for desired dispersion curve. See text for a de-scription of the dimensions.

corresponding eigenmodes [78]. Figure 8.6 shows the final design. The missing central

hole now reappears with a radius of 0.08a, and the first neighboring row is displaced

in the transverse direction towards the center by 0.15a, with an increased hole radius

to 0.325a from 0.3a.

8.3.2 Cavity design

For the cavity design problem, we found that implementing the additional constraints

(Eqn. (5.1)) in one step also takes us out of the valid eigenfunction space. However,

unlike the waveguide design, there are no further constraints to relax. We have al-

ready allowed the frequency to take on any value within the bandgap. Once again,

the problem is that the target mode (given the additional constraints) is not valid.

It is clear that as we add more stringent constraints, the hypervolume of valid eigen-

mode space decreases. What was a valid mode becomes invalid as we lose access to

the particular dielectric function. As we discussed in section 7.3.1, what we would

like to do when our desired mode is invalid is to find a mode ‘nearby’ that is a valid

one. We mentioned that the residual norm makes this a non-optimal process, but

if we are ‘close enough’, then we increase our probability of finding the correct one.

This is accomplished by iteratively ramping up to the final design constraint, rather

114

than requiring in one step ηD > ηB. At each iteration, we decrease the maximum

allowed value of ηB. We use two indicators to guide how large a stepsize we take.

First is the magnitude of the residual norm, since we know when it is large that the

inverse problem has deviated significantly from the eigenvalue problem. The second

is by comparing the predicted eigenfrequency (as given by the solution to the inverse

problem) to the actual eigenfrequency (as calculated by the solution to the subsequent

forward problem). We reduce the step size accordingly. There is no proof of conver-

gence to this adaptive scheme, but we have not found a finer mesh to produce better

results with the extra iterations. The algorithm of our approach is outlined as follows:

1. Use target mode to formulate A and b.

2. Update constraint ηB < ξi, ξi ≡ ξi−1 − δ.

3. Solve the inverse problem. If the residual norm is greater than some threshold,

decrease δ and backtrack.

4. Solve the forward problem using the result from the inverse problem.

5. Look for the eigenmode of interest. The frequency of the mode should be close

to the frequency obtained by the solver. If the frequency error is greater than

some threshold, decrease δ.

6. Use the new eigenmode as the new target mode. Record the overlap of the new

eigenmode with the original eigenmode.

7. Repeat until design constraint satisfied.

Using the iterative approach, the ‘next nearest neighbor’ remains available, whereas in

a single step approach, many neighbors simultaneously become invalid. The residual

norm metric performs worse the farther away we need to go. Using this approach, we

find a structure that supports a mode with a 93.6% overlap with the original mode

while satisfying our donor mode regime constraint (see figure 8.7).

115

Figure 8.7: Magnetic field intensity of (a) original eigenmode and (b) eigenmodesupported by final design.

Figure 8.8 shows the final design, and the first most noticeable feature is that

the bulk holes now have a reduced value of η closer to 0.7, and in some regions it

is down to 0.5. Additionally, the size of these features are also noticeably smaller.

Six of the eight nearest holes from the center now take on a ‘crescent moon’ shape

so it looks as though there is an effective lateral displacement for those holes, while

the other two remain the same. For this structure, the final average bulk dielectric

value ηB is 0.1819, significantly reduced from the initial ηB of 0.3962. The unaffected

holes correspond to regions where the field intensity remains high, which is consistent

with our intuition. The obtained eigenmode shows some leakage in the bulk area, but

otherwise clearly resembles our target mode.

The most surprising feature of this iterative method is that the performance (as

defined by the overlap integral of the obtained mode and the original target mode) was

not a monotonically decreasing function (see figure 8.9). If we examine the algorithm

closer, we see that after the first iteration, the original target mode no longer enters

the picture, at least not directly. Whatever the inverse problem solution gives us,

that becomes our next target mode. Granted, we use the mode that most resembles

our target, but otherwise it does not play a role. Considering that we would have

missed out on obtaining this solution if we had stopped a few iterations earlier, this

116

Figure 8.8: Final nanocavity design for acceptor eigenmode in donor configuration.

Iteration Number

Pe

rfo

rma

nce

(O

verlap

wit

h t

arg

et

mod

e)

Figure 8.9: Plot of the performance at each intermediate iterate of the design process.Notice that this function is not monotonic, which also provides a glimpse into thetopology of an objective function in design space.

result is somewhat troubling. However, it does highlight the difference between our

methodology and the other inverse problem approaches that use gradient type schemes

[76, 77], where there would have been no way of arriving at this design. It also reveals

that in general, the performance function is not a simple functional of the dielectric,

demonstrating real limitations of gradient based algorithms given the topology we

found. This result further highlights the need to develop a better alternative to the

residual norm as a metric for solving the inverse problem.

117

8.4 Concluding Remarks

We have set out to find a general turn-key methodology that allows arbitrary (and

by extension optimal) design of desired PBG structures. Despite the limitations

due to the convergence issues of the model, we retained the ab initio approach to

elucidate fundamental and ubiquitous barriers to optimal PBG device design due to

the topology of the space of valid eigenmodes. Having rigorously shown the non-

existence of a solution, we attempted to look for designs that best approximates the

desired functionality. We showed that the residual norm metric prevents a claim

of finding the optimal ‘next-best’ design. While other inverse problem methods,

particularly GA’s and level set methods, can more readily yield designs that can be

fabricated, they are confined to local improvements to existing design. Our approach,

though currently non-optimal, can be used to provide these local methods with new

starting points that lie ‘closer’ to the optimal design.

Even if the residual norm metric issue is worked out, our method still does not

replace these other approaches because of the convergence issues, and the problem

scales poorly with increased dimension. We were unable to take advantage of the

ideas for fast convergence [4] because we do not know a priori where the boundary

between high and low index materials will be in order to assign a tensorial value in

that discretized element. However, we can make use of gradient based methods to

go from an optimal continuous function design to the requisite binary valued design

while minimizing sacrifice in performance.

Despite these general difficulties, we were able to extend our method to obtain

remarkably good results that would be difficult to obtain with any other scheme. As

we have shown, the topology of the problem is such that a gradient based algorithm

would have missed our final design. The work we have developed here thus represents

an important addition to the inverse problem based toolbox of design methods for

the photonic problem.

118

Appendices

119

Appendix A

Sample Matlab Code to IllustrateIll-Conditioning

%Simple Matlab code to illustrate ill-conditioning

%with a numerical example

%-------------------

%

%Set Parameters

dim = 5; %sets dimension of the matrix

numiter = 10000; %number of iterations to run through

noiselevel =1e-3; %scales the magnitude of the random noise vector

%

%--------------------

%

%Define the matrix and input vector

A=1./([1:dim]’*ones(1,dim)+ones(dim,1)*[0:1:dim-1]);

%This defines the Hilbert matrix of size dimXdim

xin = ones(dim,1);

xout = A*xin;

%

%-----------------

%

120

%Statistics for the ’forward problem’

for kk = 1:numiter

noisein = noiselevel*randn(dim,1);

ein = norm(noisein)./norm(xin);

eout = norm(A*(xin+noisein)-xout)./norm(xout);

S(kk) = eout./ein;

end

figure(1);hist(S,50);

%histogram of the stability of the forward problem

%

%Statistics for the ’inverse problem’

%----------------

%Inverse problem is : Solve xin = inv(A)*xout

%

invA = inv(A);

for kk = 1:numiter

noisein = noiselevel*randn(dim,1);

ein = norm(noisein)./norm(xout);

%Note that the input and output are reversed...

eout = norm(invA*(xout+noisein)-xin)./norm(xin);

S(kk) = eout./ein;

end

figure(2);hist(S,50);

%histogram of the stability of the inverse problem

xout_round = round(1e3*xout)./1e3;

xin_bad = invA*xout_round %Shows instability to rounding

121

Appendix B

Barrier Functions of ComplexVariables

Consider the following barrier function φ(ξ) : Cn →R

φ(ξ) = −log(−f(ξ)) (B.1)

= − log

(−ξ†η + η†ξ

2

)(B.2)

where f : Cn → R, ξ ∈ Cn, η ∈ Cn. We use our Cn → R2n transformation (eqn.

(4.37)) and re-express in terms of the new variables.

[ξ] → <(ξ)

=(ξ)

≡ x (B.3)

[η] → <(η)

=(η)

≡ y (B.4)

122

where (x, y) ∈ R2n. Transforming to R2n and using eqn. (4.60) and (4.61), we find

φ(x) = − log(−g(x)) = − log(−xT y) (B.5)

∇g(x) = y (B.6)

∇φ(x) =1

xT yy (B.7)

∇2φ(x) =1

(xT y)2∇g(x)∇g(x)T (B.8)

=1

(xT y)2yyT (B.9)

Let (xT y)−2 ≡ α ∈ R, since it is just a scalar. The Hessian can be expressed as:

∇2φ = αyyT (B.10)

= α

ηr

ηi

[ηT

r ηTi ] (B.11)

= α

ηrη

Tr ηrη

Ti

ηiηTr ηiη

Ti

(B.12)

6= α

<(H) −=(H)

=(H) <(H)

(B.13)

In the final step, we show that this Hessian matrix is not equivalent to any H ∈ Cn×n

based on the Cn → R2n transformation rules for matrices (eqn. (4.47)). Therefore,

regardless of how we define the generalized complex gradient or Hessians, we cannot

include this form of barrier function in the complex domain using normal matrix

manipulation rules. We are forced to perform the cumbersome transformation to real

variables, despite suggestions found elsewhere [83, 25].

123

Appendix C

Fourier Transforms

In chapter 2, we examined the Helmholtz equation in the plane wave basis, and found

that transforming into the Fourier domain played a central role. We also alluded to

some problems associated with the truncation to finite bandwidth. In this chapter, we

look more closely at Fourier transforms, and in particular numerical implementations

of the Fourier transform as it applies to physics applications.

Fourier analysis is included in most undergraduate level curriculum, so it may seem

strange to have it included in the body of the thesis. The use of Fourier transforms,

both the continuous and discrete forms are ubiquitous in physics and engineering.

They are used extensively in signal processing, image processing and compression,

pattern recognition, and solutions of PDEs using spectral methods (as we use them

here). Most computational software such as Matlab, Mathematica, and Maple all have

built-in functions to compute these transforms. However, perhaps because it is so

commonplace, there is greater risk that one does not first think more carefully about

the underlying physics and will just let the machine grind through the calculation.

The only warning I remember as an undergrad in learning about Fourier analysis

was ‘aliasing’, and we were simply told to make sure we sample above the Nyquist

frequency. However, that is not a viable option with photonic crystals, because the

discontinuities imply an infinite Nyquist frequency. What exactly happens to this

‘aliasing’ behavior then? It turns out that there are other issues with numerical

implementations of the Fourier transform, particularly for those who wish to use

discrete Fourier transforms in computational physics. The goal of this chapter is

124

to bring awareness to some of these issues exacerbated by these discontinuities, and

justify interpreting our results in the smooth function limit which we presented in

Part II.

Organization

We will begin in section C.1 with some general bookkeeping by introducing the no-

tation we will follow in this chapter for Fourier transforms, and then revisiting the

Born von-Karman boundary conditions by paying closer attention to the topology of

the direct space and Fourier space. In section C.2 we introduce formally the discrete

Fourier transform, and in particular examine the fast Fourier transform (FFT) and

some of its properties. There are some aspects of the FFT that are unnatural to

physics applications (where the origin is usually at the center of the spectrum). This

will lead to a discussion of symmetries in section C.3, where we see real discrepancies

between what we naıvely think we are modeling, as opposed to what the mathemat-

ics is actually modeling. We also find a fundamental conflict in the symmetry of

the physics and the symmetry as a result of the discretization. In light of the issues

revealed in these sections, we emphasize the fundamental importance of considering

the continuous function limit and explore the behavior for various transform schemes.

As mentioned in section 2.4, we describe Li’s insight into the convergence issues in

the plane wave Helmholtz equation with Fourier factorization rules in section C.4,

followed by concluding remarks in section C.5.

C.1 Boundary Conditions and Fourier Space

Consider a continuous function f(r). We define the Fourier transform of the function

as

F (k) =1

V

∫

V

f(r)e−ik·rdr (C.1)

125

where V denotes the relevant integration volume. We can similarly define an inverse

Fourier transform as

f(r) =

∫

Vk

F (k)e+ik·rdk (C.2)

There are many conventions for where to place the normalization factor 1V

, and physi-

cists tend to prefer the symmetrized form, but we have chosen to follow the convention

[3] used in the treatment of the photonic bands problem. In the absence of any period-

icity that extends to infinity, the Fourier transform of an arbitrary real-space function

will be a continuous function in Fourier space. When we impose the Born von-Karman

periodic boundary conditions, as we saw in section 2.2, we are left with a discrete set

of allowed k-vectors {k}. More precisely, we have F (k′) ≡ 0 for any k′ /∈ {k}. We

mentioned that imposing periodic boundary conditions is a standard approximation,

and is considered valid in the tight-binding approximation. We explicitly examine

the difference with the following example in 1D. Consider the rectangular function

f1(x) =

A if |x| ≤ a,

0 if |x| > a.

(C.3)

The Fourier transform of f1(x) is:

F1(k) =

∫ +∞

−∞f1(x)e−ikxdx (C.4)

= 2Asin(ak)

k(C.5)

where we have ignored the normalization in eqn. (C.1) to facilitate comparison with

the periodic case.

If we now introduce an artificial periodicity by defining a supercell of length 10×2a,

126

-30 -20 -10 0 10 20 30-0.2

0

0.2

0.4

0.6

0.8

1

(a)

-1.5 -1 -0.5 0 0.5 1 1.5-0.2

0

0.2

0.4

0.6

0.8

1

(b)

Figure C.1: In (a) we show f1(x) in black, and f2(x) with the artificial periodicboundary conditions in red. The dashed line helps visualize the periodicity of f2. In(b), we plot the Fourier transform of f1 and f2. F1 is plotted in green and the allowedk values of F2 are shown as red circles. We plot in the blue curve the underlyingcontinuous form of F2 in k space. The ‘delta function train’ as a result of the real-space periodicity is shown explicitly this way.

we can define

f2(x) =

A if |x| ≤ a,

0 if a < |x| < 5a.

(C.6)

f2(x + Nxp) = f2(x) (C.7)

where N is an integer, and xp ≡ 10a is the periodicity introduced. The Fourier

transform of this function has the same form as F1(k) with the exception that it is

modulated with a delta function ‘comb’ corresponding to the allowed k values. In

figure C.1, we plot the two functions in real space and Fourier space.

The ratio of the two normalization factors we omitted accounts for the extra fea-

tures in f2(x) that are absent in the original function f1(x). However, for applications

such as spectral analysis, this is unimportant since it is the relative magnitude of the

different frequency components that matter. Quantities involving continuous k-space

integrals are well approximated by Riemann sums. We see in figure C.1 that we do

not lose information by incorporating the artificial boundary condition. Particularly

for functions like f1(x) that are negligible outside of the supercell (in our case the

function is identically zero), we see that this is a very good approximation. One of

127

our Fourier transform pair becomes a discrete sum rather than an integral:

F (k) =1

Vsc

∫

Vsc

f(r)e−ik·rdr (C.8)

f(r) =∑

k

F (k)e+ik·r∆k (C.9)

where Vsc indicates the volume of the supercell.

C.2 Numerical Implementation

With arbitrary functions, we often cannot perform the integration analytically. Nu-

merical implementation of the Fourier transform takes the form of a discrete Fourier

transform (DFT), and one of the most common algorithms for its implementation

is the fast Fourier transform (FFT). In a FFT, the real space function is sampled

at regular intervals, and the number of plane waves is truncated to allow numerical

evaluation of the transform. The number of points used in the real-space sampling

N matches the number of plane waves.

With the identification from f(x) to fx, the FFT relations are defined to be:

Fk =N−1∑x=0

fx exp

(−2πi

Nkx

), Forward FFT

fx =1

N

N−1∑

k=0

Fk exp

(2πi

Nkx

), Inverse FFT

(C.10)

where k and x are now integer values from 0 to N − 1 in units of their respective

basis vectors. In general 3D form, let {s1, s2, s3} define the periodicity of the lattice.

The unit vectors in real-space becomes {x1, x2, x3} ≡ {s1/N1, s2/N2, s3/N3} where

N1, N2, N3 are the number of sampling points along each direction. The basis vectors

128

in k-space become:

b1 = 2πs2 × s3

s1 · (s2 × s3)(C.11)

b2 = 2πs3 × s1

s2 · (s3 × s1)(C.12)

b3 = 2πs1 × s2

s3 · (s1 × s2)(C.13)

Observe as the real-space periodicity grows (|si| → ∞), the resolution increases in

k-space (i.e. |bi| → 0). The high frequency cutoff is precisely at Nyquist when we use

the same number of points in both k-space and real-space.

One reason for its popularity is that the computational complexity for the FFT

scales as O(N log N) due to the Cooley-Tukey algorithm [84], compared to O(N2)

for a direct evaluation of the sum. In creating the MPB package, Steven Johnson et

al. have also put out a free package [85] for the FFT called FFTW (Fastest Fourier

Transform in the West).

There are some notable properties about the FFT we should highlight. First, the

normalization factor is absorbed in the inverse transform, which is different than the

convention we have adopted in the continuous case. This unfortunate discrepancy

means we have to be more careful with our bookkeeping of the normalization factors,

but otherwise poses no problems. The other property which usually affects physicists

is that the FFT convention defines the domain of x and k such that the 0 value is

the first element rather than at the center of the spectrum (see eqn. (C.10)).

In physics however, the origin is usually defined at the center of most problems

that we analyze to more easily exploit symmetries. Computing the FFT with any

standard software package using discretized data in ‘physics order’ will give incorrect

results. In figure C.2 we show graphically what is effectively the conventional FFT

ordering. The correct function f(x) one needs to use in order to get what physicists

think they should get has the ‘negative’ half of the data translated by the periodicity

imposed. This translation of the spectrum in both k-space and real space is simply

a substitution of x → x + N and k → k + N respectively. In eqn. (C.10), this term

129

-15 -10 -5 0 5 10 15-0.2

0

0.2

0.4

0.6

0.8

1

-5 0 5 10-0.5

0

0.5

1

1.5Negative part of the spectrum trNegative part of the spectrum translated by 10 to the rightanslated by anslated by Negative part of the spectrum tr

(a) (b)

Figure C.2: In (a) we again show f1(x) in black, and f2(x) in red. The blue dash-dotted line defines the new domain (shaded in turquoise) of x under the FFT conven-tion. To use a standard FFT routine, we need to use the green curve rather than theblack curve to get the right results. On the right, (b) shows the equivalent k-spacedomain for the Fourier transform. The k < 0 part of the spectrum is translated overas shown. The turquoise region shows k-space domain under the FFT convention.

appears in the exponential and is evaluated to 1, so we see that the resulting FFT

is left unchanged by this translation. To be completely transparent, this means that

for a given discretized fx, if we attempt to compute the FFT coefficient Fk+N , we get

identically Fk. The generalization to 2D is illustrated graphically in figure C.3.

By defining the computational grid in FFT order rather than in physics order,

we can take advantage of the FFT algorithm without shifting the indices around. In

matlab, there are built-in functions that do this called fftshift, which go from FFT

order to physics order, and ifftshift, which go from physics order to FFT order. If the

FFT is used extensively, it is more convenient to simply define the grid in the FFT

order, and shift only when plotting the results.

This shift property is reminiscent of the concept of the reduced zone scheme

of k-space representation in solid state physics (see Chapter 9 in [12]) for periodic

potentials. However, we will see in the following section that the FFT symmetry is

artificial and one needs to be very careful how one treats these terms in the context

of computational physics.

130

5 0 5 106

4

2

0

2

4

6

8

10

Figure C.3: The four quadrants of a 2D FFT are shifted from physics ordering to FFTordering as shown. The black dash-dotted line indicates the usual physics domain,while the magenta dash-dotted line indicates the nominal FFT domain.

C.3 The Symmetry Problem

Suppose we define a 1D computational grid with N = 2n + 1 points such that

{k} = {−n,−n + 1,−n + 2, . . . , n − 2, n − 1, n}, and position {x} = {−n,−n +

1,−n + 2, . . . , n− 2, n− 1, n}. For simplicity, consider the Helmholtz operator (eqn.

(2.28)) in 1D, and notice the terms of the form ηk−k′ . This has the form of a Toeplitz

matrix (i.e. matrices of the form Am,n = Am−n), which has a natural connection

mathematically with the FFT [86]. The Toeplitz matrix has elements like ηκ, where

n < |κ| ≤ 2n, exceeding our defined k-point domain. For simple geometries, one

could use the expression for the analytical Fourier transform. For more complicated

geometries, we might be tempted to use the FFT symmetry and equate ηκ = ηκ±n

(depending on the sign of κ), since, as we saw earlier, they are formally equivalent

mathematically. This would give us the circulant form of the Toeplitz matrix. How-

131

ever, we presumably have truncated the grid at the chosen size because the omitted

high frequency components are ‘small enough’, whereas these ηκ terms can be quite

large. So how do we resolve the discrepancy on how to handle these Fourier coefficients

that lie outside our computational domain?

C.3.1 The underlying real-space function

The key is in recognizing that the real-space function is fundamental because it is

the one that corresponds to a physical quantity. With regards to the FFT symmetry,

recall that the valid but artificial periodicity in the real-space function gave rise to

the delta-function lattice in k-space. We justified the approximation by invoking

the tight-binding approximation. If we now insist on an artificial periodicity in k-

space, then necessarily (by the symmetric nature of the Fourier transform) we enforce

a convolution of any real-space function with a delta-function lattice as well. This

means that our model of the continuous function is identically zero everywhere except

the points we happen to be sampling (see figure C.4a).

This illustrates that we must not use the FFT symmetry to determine the correct

coefficients for terms outside of our computational domain. We could (as we might

with the analytical Fourier coefficients) do an oversampled transform with N × 2nD ,

where nD is the number of dimensions in real-space to evaluate those coefficients. The

problem with this approach is that it is no longer self-consistent with our truncation

condition. When we chose the computational domain, for better or for worse, we

effectively set a priori any Fourier component exceeding kmax equal to zero. Since

other quantities have the same bandwidth limitation, self-consistent treatment implies

setting those εk equal to zero as well.

Given the analysis here, it is now obvious that the underlying real-space function

for some set of FFT coefficients can be quite different than what we think they repre-

sent. Given our strict truncation approximation, we now examine the 1D rectangular

function again. This time, we take an N point FFT of the rectangular function, then

transform back to real-space with a 2N point FFT by explicitly zero padding the

132

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

0

0.5

1

1.5

(a)

(b)

Figure C.4: (a) The underlying continuous function of a nominal rectangular functionallowing the FFT symmetry. (b) The underlying continuous function of a nominalrectangular function reconstructed from the N FFT coefficients with hard truncation.

k-components outside the original bandwidth. Figure C.4b shows the true represen-

tation of an FFT. Even though the FFT/IFFT pair seems to perfectly reconstruct

a discontinuous jump, an examination of the underlying continuous function shows

the misrepresentative sampling that actually happens. Note also that after the hard

truncation we no longer have a Toeplitz matrix because of our FFT preferred order-

ing, which is different from the other works that use the PWE method and keep ηk−k′

(or the equivalent 1ε

term) as a Toeplitz matrix[11, 14, 87, 88, 20].

C.3.2 Even vs. odd

A final problem with the FFT symmetry that is quite subtle shows up in how we

choose to discretize the grid in k-space. As before, the spacing is strictly determined

by the real-space periodic BCs, while kmax is chosen according to some truncation

133

condition. The final detail addresses the boundary of the k-space grid, meaning a

determination of whether an even or an odd number of k-points are used along each

k direction. It is known that the FFT algorithm is most efficient when N is a power

of 2 or a product of small prime numbers [85]. Usual implementations choose N to

be a power of 2, which is clearly even. Here, I argue that an odd-numbered grid is

the correct choice for self-consistency considerations, particularly when we have high

frequency components we have had to truncate out.

Consider now in 1D an No point transform where No = 2n+1 and an Ne transform

where Ne = 2n of the same periodic continuous function in real-space. Shifting back

into a physics preferred coordinate system, our k-space will have in the odd case

{ko} = {−n,−n + 1, . . . , n − 1, n}. In the even case, we have a choice of either

{ke} = {−n,−n + 1, . . . , n − 2, n − 1} or {ke} = {−n + 1,−n + 2, . . . , n − 1, n},and the two are equivalent because of the FFT symmetry, i.e. any F−n = FN−n.

Consider further a real-valued real-space function (such as a dielectric function). The

continuous Fourier transform symmetry in k-space is:

Fk = F ∗−k (C.14)

This illustrates that the physical symmetry conflicts with the FFT symmetry which

arises as a result of discretizing. We do not have this conflict when we choose an odd

numbered FFT, since Fn and F−n are independent. With an even-numbered FFT of

a real-valued function, this constrains the boundary k elements to take on only real

values. In 1D, this may not be significant, since there is only one element. In 2D,

the number of boundary elements increase, and we show in figure C.5 the conflict in

symmetry.

For well-behaved functions where our sampling rate is well above Nyquist, this

conflict is not significant, simply because Fn is near zero. If we do not sample at a

sufficient rate (e.g. when we have discontinuities), the conflict becomes much more

significant. In figure C.6 we show an arbitrary discontinuous function in 2D that

can be a possible PBG dielectric function. We take the 2D FFT using an N × N

134

-4 -2 0 2 4-4

-3

-2

-1

0

1

2

3

4

FFT symmetry

Physics symmetry

Figure C.5: 2D k-space diagram showing the effect of the FFT symmetry on Fouriercoefficients at the boundaries

grid for N even and N odd. We show the resulting spectra of coefficients in figure

C.7. Our discretization choice affects not only the boundary values, but as shown

in the plot, there is a significant discrepancy between the two schemes in even the

largest of the coefficients (i.e. where the k-vector is close to the origin, far away from

the truncation limit). Since the odd numbered grid preserves the proper number

of degrees of freedom and symmetries, we consider it the discretization scheme that

is actually more appropriate for computational physics, especially when modeling

discontinuities.

C.4 Fourier Factorization

A final remark we will make about Fourier transforms as it applies to the PBG

problem deals with the Fourier coefficients of a product of two functions. Consider a

135

Figure C.6: A 2D real-space function with discontinuities that is representative of anarbitrary PBG dielectric function.

function f(x) = g(x) · h(x). The problem is to find the Fourier coefficients Fk, given

Gk and Hk, the Fourier coefficients of g(x) and h(x) respectively. This can be done

using Laurent’s Rule such that:

Fn =kmax∑

m=kmin

Gn−mHm (C.15)

where Gn−m is the familiar Toeplitz matrix. However, suppose g(x) and h(x) are

functions with concurrent discontinuities at xd, and suppose further that the discon-

tinuities at xd are complementary such that f(x) is continuous at xd, i.e.

limx→x+

d

f(x) = limx→x−d

f(x) = f(xd) (C.16)

We encounter this situation with our source free non-magnetic geometry. The mag-

netic fields are continuous everywhere, which implies (by eqn. (2.3)) that D is contin-

uous. Since ε(r) has discontinuities, E(r) must have complementary and concurrent

discontinuities such that their product is continuous.

Li proved [18] that even as we take the set {k} to infinity, Laurent’s rule will never

136

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Figure C.7: The value of the Fourier coefficients for an even numbered FFT vs. anodd numbered FFT are plotted on the complex plane, i.e. the axes are the real andimaginary parts of Fk of the function shown in figure C.6. Notice the discrepancyin most of the points. We have plotted the 81 k-points with the largest magnitude.Note that the bandwidth for both geometries are the except for the boundary point,showing the two schemes are not self-consistent.

converge. Instead, he applied what is now known as the Inverse Rule:

Fn =kmax∑

m=kmin

[Γn−m]−1 Hm (C.17)

where Γk is the Fourier expansion of γ(x) = 1g(x)

. The superscript −1 denotes matrix

inversion. The difference in convergence is shown in figure C.8.

Given Li’s analysis, it is no longer surprising why the convergence of the PWE

method depends on the polarization of the field and how we treat the dielectric

Toeplitz matrix. Rigorous application of his factorization rules to 2D and 3D has

spawned the fast Fourier factorization methods (cf. [87, 20]), but the physics is ulti-

mately embodied by the effective medium approach [17, 4].

137

(a) (b)

(c) (d)

Figure C.8: Illustration of Laurent’s rule and the inverse rule. (a) Given the Fouriercoefficients of two functions g(x) in blue and h(x) in green, we want to find the productf(x) in red. Notice a single concurrent and complementary discontinuity at x = 0,and a concurrent but non-complimentary discontinuity at x = −1. (b)Result usingLaurent’s rule. Notice the non-concurrent discontinuity had no trouble converging.(c) Magnification of problem area (d)Good convergence of result using Li’s inverserule.

C.5 Conclusion

In this appendix, we explored some of the intricacies of the numerical implementation

of Fourier transforms. For functions with discontinuities, our inherent inability to

satisfy the Nyquist criterion exacerbates the issues discussed in this chapter. For these

reasons, we cautioned against blind acceptance of the FFT/IFFT output, but instead

return to a continuous function limit interpretation. Given the accuracy limitations

revealed in chapter 2 and this appendix, self-consistency in modeling should be the

primary focus. We therefore give up on the notion of accurately modeling dielectric

slabs with etched air holes. In chapters 5–8 where we describe geometries of structures,

we use language such as a nominal geometry consisting of a lattice of air holes of

radius 0.3a and so forth. It is to be understood that we are really talking about the

real-space continuous function representation of the truncated Fourier series.

138

Appendix D

Detailed Errata for Geremia 2002

In this appendix, I will give a detailed account of the various flaws and errors in the

original photonic inverse problem paper [67] published in Physical Review E in 2002.

I feel strongly that these errors should be documented somewhere, and of course to

the extent possible, I have corrected these in the work I have done subsequent to this

paper. Unfortunately, as I was not an author in the original paper, I did not feel

it was my place to publish an errata, nor did Dr. Geremia seem motivated to do so

when I approached him with some of the initial errors. Nevertheless, if and when

the decision is made to do so, this appendix provides a sufficiently detailed account

that should be more than adequate for that purpose. Unless otherwise specified in

this appendix, reference to equation and figure numbers are meant to be for those in

Geremia 2002, while equation numbers beginning with the letter D refer to equations

in this appendix.

D.1 Introduction

D.1.1 Relevant abstract of Geremia 2002

In Geremia 2002, optimal photonic crystal cavity design results were presented using

an analytical 2D model, as well as a numerical 3D model. To date, there has not

been further investigation on the 3D work, but the 2D results have been examined

extensively. To summarize the work in Geremia 2002, the 2D analytical approach can

139

be separated into two distinct and separate steps. The first step is an optimization

of the desired mode without consideration as to the dielectric that can generate the

mode. The particular optimization that it claims to do is a maximization of pseudo-Q

factor (Q) and electric field at the origin(E(0)), and a minimization of mode volume

(V ). Quantitatively, this means a maximization of βQQ+βEE(0)−βV V , where the β’s

balance the importance of the various terms. Without loss of generality, βQ is taken

to be 1. The second step extracts the dielectric required to produce the optimized

mode from step 1 by solving the inverse Helmholtz equation. The inverse Helmholtz

equation is derived in the bulk mode basis into a set of linear equations. A ‘radially

symmetric’ defect and an asymmetric defect design were provided as illustrations to

the technique.

There are various errors or omissions in the description of the first step, the

optimization of the desired mode. We will examine these in section D.2. The more

critical error in the paper is in the derivation of the inverse Helmholtz equation

which we will address in section D.3, and of course the most notable omission is

the discussion of regularization. Before we can even address these errors though, it

turns out there are some typographical errors in the equations. In order to make a

comparison between the corrected version of the inverse Helmholtz equation and the

ones in Geremia 2002, we first need to correct the typos. We will first assume that

the derivation is correct and fix the typos in section D.1.2.

D.1.2 Typographical errors

The typos appear in equation 25 in the main text, and then A2, A3 and A4 in the

appendix. Again, actual flaws in the reasoning will be addressed later. Following the

reasoning outlined in the paper, we write out explicitly all the steps in full. We include

it all here because the derivation is extremely cumbersome, particularly keeping track

of all the indices. Of course, since the inverse problem is unstable, if you don’t know

about regularization, then even the correct equations will give bad answers, so at a

practical level when trying to code up the equations, it was rather difficult trying to

140

catch these errors. Also keep in mind that these equations will ultimately be proven

incorrect. The reader not interested in the grunge here can safely skip to section

D.1.3 for the summary of the typos on page 144.

Detailed Steps of PRE Derivation

Recall the defect mode is expanded in the TE bulk mode basis

Hm(r) =1

N

∑

n,k

a(m)n,kHn,k(r) (D.1)

where the bulk modes are computed in the plane wave expansion method:

Hn,k(r) = z∑G

hn,k+Gei(k+G)·r (D.2)

∫

VN

H∗n′,k′(r)Hn,k(r)dr = Nδn,n′

∑G

δk′,k+G (D.3)

The z vector will be omitted in the following notation for compactness unless re-

quired for completeness under curl operations. We start with Maxwell’s equation,

but separate out the defect dielectric from the unperturbed lattice:

∇× η0(r)∇×Hm(r) +∇× δη(r)∇×Hm(r) =ω2

m

c2Hm(r) (D.4)

∇× η0(r)∇×Hm(r)− ω2m

c2Hm(r) = −∇× δη(r)∇×Hm(r) (D.5)

Next, substitute into equation (D.5) the defect and bulk mode expansion of equa-

tions (D.1) and (D.2), left multiply by Hn′′,k′′ and integrate over the size of the

supercell. The left hand side (LHS) of equation (D.5) becomes

1

N

∑

n,k

a(m)n,k

ω2n,k − ω2

m

c2

∫H∗

n′′,k′′(r)Hn,k(r)dr

Evaluating the right hand side (RHS) of equation (D.5) requires a (truncated) Fourier

141

expansion of the defect dielectric.

δη(r) ≡∑

k

δηkeik·r

The k points used in the summation are the ones consistent with the specified Born-

von Karman boundary conditions (i.e. geometry of the supercell), and the series is

truncated with a finite number of reciprocal lattice vectors used to tile the reciprocal

superlattice, identical to the truncation in calculating the band structure of the bulk

lattice.

The RHS now becomes:

−∇×(∑

k′δηk′e

ik′·r)∇× 1

N

∑

n,k

a(m)n,kHn,k(r)

= − 1

N

∑

k′

∑

n,k

an,kδηk′∇× eik′·r(∇× z

∑G

hn,k+Gei(k+G)·r)

= − 1

N

∑

k′

∑

n,k

∑G

an,kδηk′hn,k+G∇× eik′·r (i(k + G)× zei(k+G)·r)

= − 1

N

∑

k′

∑

n,k

∑G

an,kδηk′hn,k+G∇× ei(k′+k+G)·r (i(k + G)× z)

= − 1

N

∑

k′

∑

n,k

∑G

an,kδηk′hn,k+G[i(k′ + k + G)]× [i(k + G)× z]ei(k′+k+G)·r

= − 1

N

∑

k′

∑

n,k

∑G

an,kδηk′hn,k+G(k′ + k + G) · (k + G)ei(k′+k+G)·rz

Next we left multiply by plane wave expansion of Hn′′,k′′ and integrate over the size

142

of the supercell again. (Omitting again the z vector for compactness)

−∫ ∑

G′′h∗n′′,k′′+G′′e−i(k′′+G′′)·r × · · ·

(1

N

∑

k′

∑

n,k

∑G

an,kδηk′(k′ + k + G) · (k + G)hn,k+Gei(k′+k+G)·r

)dr

= −∫ ∑

G′′

1

N

∑

G,k′,n,k

an,kδηk′(k′ + k + G) · (k + G)× · · ·

h∗n′′,k′′+G′′e−i(k′′+G′′)·rhn,k+Gei(k′+k+G)·rdr

= − 1

N

∑

G′′,G,k′,n,k

an,kδηk′(k′ + k + G) · (k + G)× · · ·

h∗n′′,k′′+G′′hn,k+G

∫ei(k′+k+G−k′′−G′′)·rdr

= − 1

N

∑

G′′,G,k′,n,k

a(m)n,k δηk′(k

′ + k + G) · (k + G)× · · ·

h∗n′′,k′′+G′′hn,k+G

∫ei(k′+k+G−k′′−G′′)·rdr

We have now arranged Maxwell’s equation with defect dielectric into equation

(A3) of [67]. To get to equation (A4), we simply ‘evaluate’ the integrals and collapse

the appropriate delta functions. The LHS of equation (A3) is :

1

N

∑

n,k

a(m)n,k

ω2n,k − ω2

m

c2

∫H∗

n′′,k′′(r)Hn,k(r)dr

=∑

n,k

a(m)n,k

ω2n,k − ω2

m

c2δn′′,n

∑G

δk′′,k+G , using equation (D.3)

=∑

k

a(m)n′′,k

ω2n′′,k − ω2

m

c2

∑G

δk′′,k+G

We note that by Bloch’s theorem, ωn,k = ωn,k+G, so we can in this instance substitute∑

G δk′′,k+G with Nδk′′,k. Finally, we obtain the expression

Na(m)n′′,k′′

ω2n′′,k′′ − ω2

m

c2

143

The RHS of equation (A4) simply involves collapsing the integral into a delta

function. Only note here is that since we are integrating over the supercell, we pick

up a factor of N .

∫

V

ei(k′+k+G−k′′−G′′)·rdr = NδG,k′′+G′′−k−k′

The RHS of equation (A3) after collapsing this delta function on G = k′′ + G′′ − k− k′

is

− 1

N

∑

G,G′′

∑

k′

∑

n,k

a(m)n,k δηk′(k

′ + k + G) · (k + G)× . . .

h∗n′′,k′′+G′′hn,k+G

∫ei(k′+k+G−k′′−G′′)·rdr

= −∑

G′′

∑

k′

∑

n,k

a(m)n,k δηk′(k

′′ + G′′ − k′) · (k′′ + G′′)× h∗n′′,k′′+G′′hn,k′′+G′′−k′

= −∑

G′′

∑

k′

∑

n,k

a(m)n,k h∗n′′,k′′+G′′hn,k′′+G′′−k′(k

′′ + G′′ − k′) · (k′′ + G′′)δηk′

Multiply the collapsed versions of equation (A3) by − 1N

to recover equation (A4).

Next we note as in equation (A5) that we can fold summations over k back into the

First Brillouin zone using the identity:

∑

n,k

a(m)n,k = N

∑n,q

a(m)n,q

Next we make the following index transformations: n → n′, G′′ → G, n′′ → n.

We also collapse k to q′ using the above identity, and rename k′ to k. Finally we

restrict the indices k′′ to within the first Brillouin zone, and rename it q, since they

are not summed.

144

The LHS of equation (A4) is:

1

N

∑

G′′

∑

k′

∑

n,k

a(m)n,k h∗n′′,k′′+G′′hn,k′′+G′′−k′(k

′′ + G′′ − k′) · (k′′ + G′′)δηk′

=∑G

∑

k

∑

n′,q′a

(m)n′,q′h

∗n,q+Ghn′,q+G−k(q + G− k) · (q + G)δηk

=∑

k

{∑G

∑

n′,q′a

(m)n′,q′h

∗n,q+Ghn′,q+G−k(q + G− k) · (q + G)

}δηk

≡∑

k

D(m)n,q;kδηk

Finally the RHS of equation (A4) is:

−a(m)n′′,k′′

ω2n′′,k′′ − ω2

m

c2

= a(m)n,q

ω2m − ω2

n,q

c2

This gives us equation (24) as desired, with the inversion matrix D given by

equation (25).

D.1.3 Proposed typographical errata

Equation 25 should read:

D(m)n,q;k =

∑

n′

∑

G,q′a

(m)n′q′h

∗n,q+Ghn′,q+G−k × (q + G) · (q + G− k)

rather than

D(m)n,q;k =

∑

n′

∑

G,q′a∗n′q′h

∗n,q+Ghn′,q′+G−k′ × (q + G) · (q + G− k′)

145

Equation A2 should read:

∫

VN

H∗n′,k′(r)Hn,k(r)dr = Nδn′,n

∑G

δk′,k+G

rather than ∫

VN

H∗n′,k′(r)Hn,k(r)dr = δn′,n

∑G

δk′,k+G

Equation A3 should read:

1

N

∑

n,k

a(m)n,k

ω2n,k − ω2

m

c2

∫H∗

n′′,k′′(r)Hn,k(r)dr

= − 1

N

∑

k′

∑

n,k

∑

G,G′′a

(m)n,k δηk′h

∗n′′,k′′+G′′hn,k+G · · ·

×(k′ + k + G) · (k + G)

∫ei(k′+k+G−k′′−G′′)·rdr

rather than

1

N

∑

n,k

a(m)n,k

ω2n,k − ω2

m

c2

∫H∗

n′′,k′′(r)Hn,k(r)dr

=1

N

∑

k

∑

n,k

∑

G,G′′a

(m)n,k δηkh

∗n′′,k′′+G′′hn,k+G · · ·

×(k′′ + G′′) · (k′′ + G′′ − k′)∫

ei(k′+k+G−k′′−G′′)·rdr

Equation A4 should read:

1

N

∑

n,k

∑

k′

∑

G′′a

(m)n,k δηk′h

∗n′′,k′′+G′′hn,k′′+G′′−k′(k

′′ + G′′) · (k′′ + G′′ − k′)

= −a(m)n′′,k′′

ω2n′′,k′′ − ω2

m

c2

146

rather than

1

N

∑

n,k

∑

k′

∑

G′′a

(m)n,k δηk′h

∗n′′,k′′+G′′hn,G′′−k′(k

′′ + G′′) · (k′′ + G′′ − k′)

= − 1

Na

(m)n′′,k′′

ω2n′′,k′′ − ω2

m

c2

D.2 Mode Optimization Errors

The three physical quantities that were optimized were maximizing E field intensity

at the central defect location, minimizing the mode volume, and maximizing the

Q factor. In the paper, the optimization is performed using Lagrange multipliers.

Equation (23) from the paper states:

∑

n′,q′

[ωn′,q′ωn,q

qq′+ βIH

∗n′,q′(0)Hn,q(0)− βV 〈ψn′,q′|ψn,q〉

]a

(m)n′,q′ = Λan,q

The first term maximizes Q by removing bulk mode contribution above the light-

line, the second term maximizes the intensity at the origin, and the final term min-

imizes the mode volume. As it appears in the paper, it is unclear how it actually

optimizes any of the three quantities. We will address each expression in the next

sections.

D.2.1 Q factor

The Q factor optimization incorrectly applies the lightline model to mimic the be-

havior of the cavity Q. The light cone rule of thumb originates from the idea of lossy

modes coupling into free space. The relevant traits that describe modes lying above

the light line (lossy modes) is that they have high frequencies and small in-plane

fourier components. For a defect mode of a given frequency, correct application of

this rule of thumb is to restrict fourier components lying within a small circle (sphere

in 3D) in k-space determined by the mode frequency. This is the approach consistent

with Oskar Painter’s group [9]. So there are two errors in using the frequency of the

147

associated bulk mode and its quasi k-vector in determining where it lies in relation to

the light line. The first point is that the actual frequency of the defect free bulk mode

is irrelevant, since we are trying to determine the lossiness of the defect mode. Hence

only the defect mode frequency will give the relevant coupling condition. Secondly, it

is the fourier components that compose the mode that is relevant, and not the quasi

k-vector. A bulk mode in fact is composed of the fourier components of the k-vector

in the first Brillouin zone as well as all others by addition of reciprocal lattice vectors.

In fact, many of the upper band bulk modes actually have large fourier components,

and are minimally lossy (i.e. minimal, though finite contribution from small fourier

components). A reduced zone scheme picture is actually rather deceptive in trying to

visualize the light line. On the other hand, the bulk modes originating from the low

lying bands (of which most of the ‘optimized’ modes are composed) actually have a

significant amount of small fourier components. Therefore, if one were to use these

low lying bulk modes to form a defect mode, since the defect frequency is in the band

gap, these contributions are in fact highly lossy due to the expanded region of un-

favorable k-space. In misapplying the light line constraint, the optimization should

make things worse by incorporating more of these low lying modes. Finally, this

analysis ignores the fact that coherent superposition of lossy bulk modes can lead to

cancellation of small (i.e. leaky) fourier components. Going back to our linear algebra

language, the bulk modes are a non-diagonalized basis for the purpose of Q factor

considerations. The point is moot when only using 5 layers of photonic crystal as

in [67], because there are so few small fourier components for such a small supercell

(although note the observation in section D.4 about figures 3 and 5).

D.2.2 E field intensity

In the paper, it is unclear whether it is the H field or the E field that actually gets

optimized. The equations in the paper throughout only show expressions for the H

field (cf. eqns. (5,21,23)), whereas the plots in the results section show the E field.

One can obtain the E field by taking the curl of H, but because TE polarization

148

is assumed, the E field lies in the xy-plane (as opposed to the H field which only

has z components), it must be treated as a vector quantity. The expression for the

E optimization is not discussed in the paper. One way to treat this is to allow for

vectored expansion coefficients for the optimized E, but then it is not immediately

obvious that the Lagrange multipliers method is applicable. Convex optimization

cannot be applied here either because the norm of E(0) is convex, and we cannot

maximize a convex function using convex optimization. In fact, the local extremum

of a norm must be a local minimum, so the global maximum must be some end point.

In any case, equations (23) in the paper is an incorrect expression for a cavity mode

optimization, since maximizing the H field intensity will yield a zero E field at the

origin.

D.2.3 Mode volume

There are also problems with the expression for the mode volume optimization in the

paper. Again, there is the ambiguity over whether it is the H field or E field that

is optimized (cf. eqns. (4,19,20,23)). Even if we were to assume that it is sufficient

to minimize the mode volume of H to get the desired result, the expressions are still

incorrect. In the paper, ψ refers to a max 1 normalized mode function that satisfies

Hj(r) = H0,jψj(r),

where j is some mode index, and |ψ(r)|max = 1. The mode volume for the mode j is

then

Vj =

∫|ψj(r)|2dr

The expression so far is fine. However, directly evaluating the mode volume by an

expansion as the paper attempts to do in equation (19)

Vm =∑

n′,q′

∑n,q

a(m)∗n′,q′a

(m)n,q 〈ψn′,q′(r)|ψn,q(r)〉+ c.c.

149

is in fact not so straightforward, and unfortunately incorrect. The corresponding

normalization constant H0,m for the mode m cannot be determined without first

doing the summation. Using equation (14) as the expansion for the target mode

implies:

Hm(r) =∑

n

∑q∈BZ

a(m)n,q Hn,q(r) (D.6)

= H0,m(r)ψm(r) (D.7)

Eqn. (D.7) explicitly shows equation (19) is incorrect because

ψm(r) 6=∑

n

∑q∈BZ

a(m)n,q ψn,q(r) (D.8)

A simple example that illustrates the error is to choose any basis that has all of its

basis functions max 1 normalized, such as the plane wave basis. In such a basis,

the RHS of eqn. (D.8) is identically Hm(r), which is clearly not max 1 normlized in

general (hence it cannot equal ψm(r) by definition). All equation (19) represents is

in fact∫ |H′

m(r)|2dr, for some unnormalized mode H′m, uncorrelated with the actual

mode volume. The physical meaning of this term as written in the paper is unclear,

but it does skew the optimized mode with some completely undesired effect.

As it turns out, formulating the mode volume minimization is actually unnecessary

in this case. By imposing an energy constraint to the optimization, a maximized E

field will necessarily have minimum mode volume. If we think about this carefully,

it is clear since the point of minimizing mode volume is to maximize the electric

field strength per photon. An energy constraint can be imposed by normalizing the

expansion coefficients. Obviously in the E field maximization step, this is already

done, otherwise by scaling the coefficients, we can increase E(0) without bound. In

other applications where one cannot avoid dealing with the mode volume directly, it

would be prudent to exercise more caution in its treatment.

A final comment about the optimization procedure concerns the weighting factors

(β’s). At the conclusion of Section III-C, it talks about optimizing over the βi coeffi-

150

cients, and the idea presented is to nest the inverse equation (equation (24)) within

an outer βi optimization loop. However, a performance metric for the βi’s is not pro-

vided (not even a qualitative comment on what makes a set of β better than another

set.) If it is just to maximize the I function (i.e. the total objective function shown

in equation (8)) over all combinations of β’s, then that seems to defeat the purpose

of having those coefficients in the first place. These coefficients enforce the fact that

there are tradeoffs between the different properties. I can be made large if one sets

βV = 0, effectively removing the requirement for having small mode volume. We

simply could not ascertain as to what exactly the algorithm was optimizing. Section

III-E and III-F explain that a conjugate-gradient algorithm is used for the optimiza-

tion with the inverse problem nested within it. No explanations were presented as

to why this nesting was necessary, nor how it would be accomplished. For example,

how is the solution of the inverse problem (eqn. (24)) used in the subsequent iter-

ation? One would expect that it involves evaluating the gradient of the matrix D

with respect to β somehow, but eqn. (24) depends on β implicitly through the mode

expansion coefficients an,q. Eqn. (23) shows us that the an,q’s are eigenvectors to an

eigenvalue problem parameterized in part by the β’s. The study of the rotation of

eigenvectors due to perturbations to its operator is not a simple matter [78], so it is

not obvious or clear how one might evaluate the gradient.

D.3 Inversion Errors

The second step of the design algorithm involved inverting the Helmholtz equation

using the bulk modes as a basis in the paper. From the discussion of the Q factor

optimization in section D.2, it is clear that the bulk mode basis is not a natural

basis for this problem. As we saw in chapter 6, inverting the equation in the plane

wave basis (the natural basis for Q factor type considerations) actually gives a very

simple and concise result. Treatment in the bulk mode basis was unnecessarily messy

and convoluted, further complicated by the decision to sum over multiple Brillouin

zones. Consequently, the derivation gave rise to terms that include Hn,k(r). The

151

term Hn,k(r) is actually rather awkward to interpret, since the purpose of the band

index is to reduce all k-vectors into the first Brillouin zone. It is ambiguous what is

really meant to run the k-vectors through the entire k-space and still have a band

index. Ignoring the strange notation and continuing to expand the bulk mode shows

that the assumed orthogonality relation in equation (A2)

∫

VN

H∗n′,k′(r)Hn,k(r)dr = δn,n′

∑G

δk′,k+G

is invalid and therefore misused in the derivation of the inversion equations. A casual

examination of this relation would lead us to believe that it appears correct, save

for the missing normalization factor of N for repeating the summation over multiple

Brillouin zones (see corrected equation (A2) on page 145). However, if we really try

to rigorously write down what this notation is supposed to mean, the subtle errors

turn out to be quite significant. I will use an example to illustrate.

First, let us revisit the concept of the band index and the q’s in the first Brillouin

zone. The number of bands is equivalent to the number of Brillouin zones (BZ) in

k-space, which is the same as the number G-vectors we keep. For a truncated k-

space, this implies that the total number of k-points in the computational domain

is equal to NG ×Nq, where NG is the number of G-vectors and Nq is the number of

wavevectors in the first BZ. Therefore, the label {n, q} can be mapped to k by the

relation kn,q = q+Gn. We explicitly label k with the subscripts to help us designate

where it comes from. This implies that the term

Hn,k = Hn,kn′,q

= Hn,q+Gn′

= Hn′′,q

where Gn′′ = Gn + Gn′

Of course, there are no guarantees as to whether Gn′′ is within the truncated set or

not. The reader can refer to appendix C for proper and improper ways of handling

152

G1

G2

G4

G5

G3

Figure D.1: The dotted line outlines the first Brillouin zone of the defect free bulkphotonic lattice in k-space, and the red dots are the lattice sites of the reciprocallattice. As shown above, G3 = G1 + G2 = G4 + G5.

these terms, as this is not addressed in the paper. There are problems even if we do

not exceed the truncation domain. Suppose we have a set of G such that G1 +G2 =

G4 + G5 = G3 as illustrated in figure D.1, and consider

H∗n1,k1

Hn2,k2 = H∗n1,kn′1,q

Hn2,kn′2,q= H∗

n′′1 ,qHn′′2 ,q

Let

n1 = 1, n′1 = 2

n2 = 4, n′2 = 5

∴ n′′1 = n′′2 = 3

So Hn1,k1 = Hn2,k2 , but according to equation (A2), these are orthogonal to each

other since n1 6= n2. The other scenario would be if we choose n1 = n2, but n′1 6= n′2

such that n′′1 6= n′′2. The two modes should actually be orthogonal, but the RHS of

equation (A2) would still sum to 1. Getting this orthogonality condition wrong will

lead to an incorrect set of inversion equations.

153

Of course, we have already shown that the better way to do the inversion is to

stay in the plane wave basis. However, to show that the derivation in Geremia 2002 is

indeed incorrect, we will now rederive the inversion equation in the bulk mode basis,

but done properly with the right orthogonality condition. As a final comment, it

should be pointed out that besides the fact that the plane wave basis is the natural

basis for Q factor considerations, it turns out that from a computation point of view,

formulating the inversion problem in the bulk mode basis is also significantly less

efficient. Geremia 2002 reports a N5 scaling for forming the D matrix, whereas in

the plane wave basis, forming our inversion matrix scales as a more reasonable N2.

There appears to be no apparent advantage nor any compelling reason to use the

bulk mode basis for any of this work.

D.3.1 Correct derivation in the bulk mode basis

We start with Maxwell’s equation, but separate out the defect dielectric from the

unperturbed lattice:

∇× η0(r)∇×Hm(r) +∇× δη(r)∇×Hm(r) =ω2

m

c2Hm(r) (D.9)

∇× η0(r)∇×Hm(r)− ω2m

c2Hm(r) = −∇× δη(r)∇×Hm(r) (D.10)

We recall that δη(r) and Hm(r) can be expanded as:

Hm(r) ≡∑B

aBHB(r) (D.11)

δη(r) ≡∑

k

δηkeik·r (D.12)

The aB are the expansion coefficients of the defect mode in the bulk mode basis, HB(r)

are the bulk modes of the perfect lattice, and the k points used in the summation

are the ones consistent with the specified Born-von Karman boundary conditions (i.e.

geometry of the supercell.) The series is truncated with a finite number of reciprocal

lattice vectors used to tile the reciprocal superlattice, identical to the truncation in

154

calculating the band structure of the bulk lattice. The bulk modes are computed in

the plane wave basis, and we recall the properties of the bulk modes for a defect-free

lattice. The notation that is used here is slightly different than the usual one, so we

will elaborate slightly for the sake of clarity. For a defect-free bulk photonic crystal,

the usual notation labels all modes by a band index n and a wave vector in the First

Brillouin zone q, and the bulk mode label B replaces the {n, q} notation. Bulk modes

of different bands and/or different q are orthogonal to one another. In the plane wave

expansion method, each q yields an independent NG×NG eigenvalue problem of the

form

Θqhn,q =ωn,q

c2hn,q (D.13)

HBn,q(r) ≡ Hn,q(r) =∑G

hn,q+Gei(q+G)·r (D.14)

with the components of the eigenvectors acting as the expansion coefficient for the

associated mode. In the supercell method, the Brillouin zone of the superlattice

is reduced so that it contains only a single k point, and in fact, the {k} become

the reciprocal lattice vectors (i.e. {G} in the bulk scenario) for the supercell. The

eigenvalue problem now involves all k vectors, and there is only a single q = 0. For

a defect-free lattice in the supercell description, the independence of the bulk modes

imply that the operator can be expressed in block diagonal form

Θsupercellq=0 = Θq1 ⊕ Θq2 ⊕ Θq3 ⊕ · · · ⊕ Θqn (D.15)

if the k are ordered such that

{k1, . . . ,kN,kN+1, . . . ,k2N, . . . ,kn×N}

correspond to

{q1 + G1, . . . ,q1 + GN,q2 + G1, . . . ,q2 + GN, . . . ,qn + GN}

155

Therefore, our plane wave expansion of the bulk modes will be expressed as

HB(r) =∑

k

hB,keik·r

The orthogonality relation is therefore

∫

s.c.

H∗B(r)HB′(r)dr = NδB,B′ (D.16)

where the area of the supercell is N times that of the bulk unit cell.

We left multiply equation (D.5) by H∗B′′(r) and integrate to obtain for the LHS:

∫H∗

B′′(r)

[∇× η0(r)∇×Hm(r)− ω2

m

c2Hm(r)

]dr

=

∫H∗

B′′(r)

[−ω2

m

c2+∇× η0(r)∇×

] ∑

B′aB′HB′(r)

=∑

B′aB′

(ω2

B′ − ω2m

c2

) ∫H∗

B′′HB′dr

= N

(aB′′

ω2B′′ − ω2

m

c2

)

The right hand side becomes:

−∫

H∗B′′(r) [∇× δη(r)∇×Hm(r)] dr

= −∫

H∗B′′(r)

[∑

k,B

aB∇× δηkeik·r∇×HB(r)

]dr

= −∫

H∗B′′(r)

[ ∑

k,k′,B

aB∇× δηkeik·r∇× hB,k′e

ik′·r]

dr

= −∫

H∗B′′(r)

[ ∑

k,k′,B

aBδηkhB,k′(k + k′) · (k′)eik·r]

dr

= −∑

k,k′,k′′,B

h∗B′′,k′′aBδηkhB,k′(k + k′) · (k′)∫

ei(k+k′−k′′)·rdr

= −N∑

k,k′′,B

aBδηkh∗B′′,k′′hB,k′′−k(k

′′) · (k′′ − k)

156

where on the last line we have chosen to collapse k′ onto k′′ − k.

Equation (D.10) then becomes

∑

k

(∑

k′′,B

aBh∗B′′,k′′hB,k′′−k(k′′) · (k′′ − k)

)δηk = aB′′

ω2m − ω2

B′′

c2

∑

k

DB′′,kδηk = aB′′ω2

m − ω2B′′

c2

where DB′′,k ≡∑

k′′,B

aBh∗B′′,k′′hB,k′′−k(k′′) · (k′′ − k) (D.17)

D.3.2 Compare with PRE derivation

We are finally ready to compare our results here with the previously obtained ex-

pression. We will need to translate our notation again so that a valid comparison of

the inversion matrix D can be made. The sum over the bulk modes B is equivalent

to all (n,q) pairs in the original notation, while the sum over k′′ will be converted

to a double sum over (q′,G) with k′′ = q′ + G. Therefore, equation (D.17) can be

rewritten as:

Dn′′,q′′;k′′ =∑

q′,G,q,n

anqh∗n′′q′′,q′+Ghnq,q′+G−k(q

′ + G) · (q′ + G− k) (D.18)

Relabelling the indices {n,q} → {n′,q′}, {q′ → q′′}, and {n′′,q′′} → {n,q} produces

Dn,q;k =∑

q′′,G,n′,q′an′q′h

∗nq,q′′+Ghn′q′,q′′+G−k(q

′′ + G) · (q′′ + G− k) (D.19)

6= Dn,q;k =∑

n′,G,q′an′q′h

∗nq+Ghn′,q+G−k(q + G) · (q + G− k) (D.20)

where equation (D.20) is the old expression. The q′′ does not appear in the old

expression, so we will have to look more closely at certain terms to reveal what the

index is actually doing. We note that in equation (D.19),

h∗nq,q′′+G = 0 if q 6= q′′

157

since the indices (n,q) represent the bulk mode obtained from the q eigenvalue prob-

lem, independent of all other {q′′}’s. Therefore,

h∗nq,q′′+G = h∗nq,q′′+Gδq,q′′

and the sum over q′′ in equation (D.19) collapses to produce

Dn,q;k =∑

n′,G,q′an′q′h

∗nq,q+Ghn′q′,q+G−k(q + G) · (q + G− k) (D.21)

Examining the two equations closely, the discrepancy is in the factor hn′q′,q+G−k. In

the new expression, hn′q′,q+G−k has an associated delta function that collapses the

sum over q′ to a single q0 which is the q + G− k vector translated back into the

First Brillouin zone. In the old expression, this requirement is not present. The n′

index specifies the ‘band’ of interest, and in general, the expansion coefficients for

q0 will not be zero for some given band. This is the manifestation of the incorrect

orthogonality expression.

D.3.3 Solving the inverse equation

When it comes to finally solving the inverse equation

∑

k

D(m)n,q;k′δηk = a(m)

n,q

ω2m − ω2

n,q

c2

the paper never discussed how one actually goes about solving this set of (incorrect)

linear equations. The end of Section III-D suggested using approximate methods for

solving linear systems of equations and referenced a book by Golub and Van Loan [89]

that focusses on numerical implementation of standard linear algebra routines. From

chapter 3, we now know that standard linear algebra techniques will not work because

the problem is ill-conditioned. The essential tool to use for this type of problem is

regularization, but that is somehow omitted in the paper. Regardless, we can use

the regularization technique to solve the h1 defect mode problem, as was done in

158

chapter 6, but using equation (24) from the paper. We perform our proof of principle

calculation as in section 6.4. No amount of regularization enabled the recovery of the

nominal h1 geometry, and this is without the addition of any noise term.

D.4 Results Errors

We end this appendix with a discussion of the results presented in the paper. There

is a slight misnomer with the term radially symmetric defect, since the defect intro-

duced is not radially symmetric, but rather it retains the rotational symmetry of the

hexagonal lattice. (An additive radially symmetric perturbation cannot produce a

displacement of holes surrounding the defect without changing the dielectric between

the holes at the same radius.) The details are again somewhat vague, but private

communications with the first author confirmed that it indeed was radial symmetry

that was enforced in the calculation. It was mentioned that it made the problem

easier since it essentially became a 1D problem (solve for η(|r|) rather than η(r)).

It is not clear how one can actually use this bulk mode expansion formulation (or

even PWE) to enforce radial symmetry, but rotational symmetry can in principle be

enforced by selecting only the k-vectors that preserve the desired symmetry. We will

assume that in fact, the paper meant defects with hexagonal rotation symmetry. If

it is the hexagonal symmetry that is enforced, then the target mode would need to

have been expanded in terms of the reduced set of k-vectors, and thus have hexagonal

symmetry. The mode shown in figure 1(b) does not have the correct symmetry. Of

course, working in the bulk mode basis, it is still unclear which bulk modes should

be kept to preserve that symmetry. Even in the plane wave basis, working out the

proper symmetry transformations in the inversion matrix is not trivial. That will be

left as an exercise for the reader.

To demonstrate the Q factor optimization, figure 3 (and also figure 5 for the

asymmetric defect) shows the bulk mode contribution using a dispersion diagram to

indicate significant contributors to the mode as an attempt to show the removal of

leaky modes. The number of k-points taken in the expansion along the Γ − X line

159

is much greater than 5, which is inconsistent and incompatible with the specified

supercell geometry (5 layers surrounding the central defect). There is a discussion in

Section III-D about the dimension required of the matrix D. Specifically, it claimed

that the number of G’s must equal the number of q’s, which is incorrect. Briefly, the

number of G’s dictate the real-space resolution one can achieve, while the number of

q’s is determined by the size of the supercell. One can have many layers using poor

resolution or vice versa. Appendix C explains how to discretize k-space appropriately,

given the specified boundary conditions.

The smoothed dielectric function that is given by the solution to eqn. (24) (with

the defect-free geometry added) is shown in figure 2, and according to Sec. III-E,

the nominal structure was extracted from this function by taking a contour plot

along (ηmax + ηmin)/2. According to the text and in figure 1, the holes surrounding

the central defect had a reduced radii, and were displaced radially from the original

lattice positions. However, the reduction in hole radii and radial displacement are

not observed in the 3D view of the dielectric function in figure 2. In addition, none of

the surrounding holes exceed the threshold for a valid contour as defined above. For

ηmax = 1 and ηmin > 0, this gives η > 0.5, while figure 2 shows the surrounding regions

have η less than 0.5. Again, it is hard to be quantitative given only these contour

plots, but it is still worth contemplating the following: What does δη need to look like

in order to actually get these small displacements to the hole locations (as shown in

the paper) without distorting the circular shape? And, and for the asymmetric defect,

what is the δη required to stretch and displace these circular holes into elliptical ones

so the design has these fractional edge dislocation like features?

Neither solution includes a comparison of Q, V and E(0) of the actual obtained

mode with those from the original optimized target mode. One is left to assume

that whatever solution was obtained, it perfectly gave back what was designed. We

now know that when one regularizes, the solution rarely matches exactly what you

had designed. One can only hope that the solution is ‘close’ in some relevant sense.

Unfortunately, these crucial comparisons, as well as other important details were

omitted.

160

Bibliography

[1] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge

University Press, 2004.

[2] Kane S. Yee. Numerical solution of initial boundary value problems involving

Maxwell’s Equations in isotropic media. IEEE Transactions on Antennas and

Propagation, (3):302–307, 1966.

[3] M. Plihal and A. A. Maradudin. Photonic band structure of two-dimensional

systems: The triangular lattice. Physical Review B, 44:8565–8571, 1991.

[4] Steven G. Johnson and John D. Joannopoulos. Block-iterative frequency-domain

methods for Maxwell’s equations in a planewave basis. Optics Express, 8(3):173–

190, 2001.

[5] C. T. Chan, Q. L. Yu, and K. M. Ho. Order-N spectral method for electromag-

netic waves. Physical Review B, 51(23):635–642, 1995.

[6] John D. Joannopoulos, Robert D. Meade, and Joshua N. Winn. Photonic Crys-

tals: Molding the Flow of Light. Princeton University Press, 1995.

[7] Kazuaki Sakoda. Optical Properties of Photonic Crystals. Springer Verlag Pub-

lishing, 2001.

[8] George B. Arfken and Hans J. Weber. Mathematical Methods for Physicists.

Academic Press, 5th edition, 2001.

[9] Kartik Srinivasan and Oskar Painter. Momentum space design of high-Q pho-

tonic crystal optical cavities. Optics Express, 10(15):670–684, 2002.

161

[10] K. M. Leung and Y. F. Liu. Full vector wave calculation of photonic band struc-

tures in face-centered-cubic dielectric media. Physical Review Letters, 65:2646–

2649, 1990.

[11] K. M. Ho, C. T. Chan, and C. M. Soukoulis. Existence of a photonic gap in

periodic dielectric structures. Physical Review Letters, 65:3152–3155, 1990.

[12] Neil W. Ashcroft and N. David Mermin. Solid State Physics. Saunders College

Publishing, 1976.

[13] H. S. Sozuer, J. W. Haus, and R. Inguva. Photonic bands: Convergence problems

with the plane-wave method. Physical Review B, 45(24):13962–13972, 1992.

[14] Shangping Guo and Sacharia Albin. Simple plane wave implementation for pho-

tonic crystal calculations. Optics Express, 11(2):167–175, 2003.

[15] Yongjun Cao, Zhilin Hou, and Youyan Liu. On shape optimization of optical

waveguides using inverse problem techniques. Inverse Problems, 17:1141–1162,

2001.

[16] Rossella Zoli, Marco Gnan, Davide Castaldini, Gaetano Bellanca, and Paolo

Bassi. Reformulation of the plane wave method to model photonic crystals.

Optics Express, 11(22):2905–2910, 2003.

[17] R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L.

Alerhand. Accurate theoretical analysis of photonic band-gap materials. Physical

Review B, 48(11):8434–8437, 1993.

[18] Lifeng Li. Use of fourier series in the analysis of discontinuous periodic structures.

Journal of the Optical Society of America A, 13:1870–1876, 1996.

[19] Linfang Shen and Sailing He. Analysis for the convergence problem of the plane-

wave expansion method for photonic crystals. Journal of the Optical Society of

America A, 19(5):1021–1024, 2002.

162

[20] Aurelien David, Henri Benisty, and Claude Weisbuch. Fast factorization rule

and plane-wave expansion method for two-dimensional photonic crystals with

arbitrary hole-shape. Physical Review B, 73:075107, 2006.

[21] Heinz W. Engl, Martin Hanke, and Andreas Neubauer. Regularization of inverse

problems. Kluwer Academic Publishers, 2000.

[22] Sze Tan and Colin Fox. Physics 707 – Inverse Problems. Lec-

ture notes from University of Auckland Course on Inverse Problems;

http://www.math.auckland.ac.nz/ phy707/.

[23] Per Christian Hansen. Regularization tools: A matlab package for analysis and

solution of discrete ill-posed problems. Numerical Algorithms, 6:1–35, 1994.

[24] Arnold Neumaier. Solving ill-conditioned and singular linear systems: A tutorial

on regularization. SIAM Review, 40(3):636–666, 1998.

[25] K. Petersen and M. Pedersen. Matrix Cookbook. The Matrix Cookbook: A

mathematical desktop reference on matrices; http://matrixcookbook.com.

[26] Eli Yablonovitch. Inhibited spontaneous emission in solid-state physics and elec-

tronics. Physical Review Letters, 58(20):2059–2062, 1987.

[27] E. Yablonovitch and T. J. Gmitter. Photonic band structure: The face-centered-

cubic case. Physical Review Letters, 63(18):1950–1953, Oct 1989.

[28] E. Yablonovitch, T. J. Gmitter, and K. M. Leung. Photonic band structure: The

face-centered-cubic case employing nonspherical atoms. Physical Review Letters,

67(17):2295–2298, Oct 1991.

[29] W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe,

and J. D. Joannopoulos. Measurement of photonic band structure in a two-

dimensional periodic dielectric array. Physical Review Letters, 68(13):2023–2026,

Mar 1992.

163

[30] Pierre R. Villeneuve and Michel Piche. Photonic band gaps in two-dimensional

square lattices: Square and circular rods. Physical Review B, 46(8):4973–4975,

Aug 1992.

[31] Eli Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer,

and J. D. Joannopoulos. Donor and acceptor modes in photonic band structure.

Physical Review Letters, 67(24):3380–3383, 1991.

[32] Robert D. Meade, Karl D. Brommer, Andrew M. Rappe, and J. D. Joannopou-

los. Photonic bound states in periodic dielectric materials. Physical Review B,

44(24):13772–13774, Dec 1991.

[33] M. H. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ip-

pen, and H. I. Smith. A three-dimensional optical photonic crystal with designed

point defects. Nature, 429:538–524, 2004.

[34] Steven G. Johnson and J. D. Joannopoulos. Three-dimensionally periodic dielec-

tric layered structure with omnidirectional photonic band gap. Applied Physics

Letters, 77(22):3490–3492, 2000.

[35] T. Sondergaard, A. Bjarklev, J. Arentoft, M. Kristensen, J. Erland, J. Broeng,

and S. E. Barkou Libori. Designing finite-height photonic crystal waveguides:

confinement of light and dispersion relations. Optics Communications, 194:341–

351, 2001.

[36] Steven G. Johnson, Pierre R. Villeneuve, Shanhui Fan, and J. D. Joannopoulos.

Linear waveguides in photonic-crystal slabs. Physical Review B, 62(12):8212–

8222, 2000.

[37] Ali Adibi, Yong Xu, Reginald K. Lee, Amnon Yariv, and Axel Scherer. Properties

of the slab modes in photonic crystal optical waveguides. Journal of Lightwave

Technology, 18(11):1554–1564, 2000.

164

[38] Ali Adibi, Yong Xu, Reginald Lee, Amnon Yariv, and Axel Scherer. Design of

photonic crystal optical waveguides with singlemode propagation in the photonic

bandgap. Electronic Letters, 36(16):1376–1378, 2000.

[39] Marko Loncar, Jelena Vuckovic, and Axel Scherer. Methods for controlling po-

sitions of guided modes of photonic-crystal waveguides. Journal of the Optical

Society of America B, 18(9):1362–1368, 2001.

[40] Y. Desieres, T. Benyattou, R. Orobtchouk, A. Morand, P. Benech, C. Grillet,

C. Seassal, X. Letartre, P. Rojo-Romeo, and P. Viktorovitch. Propagation losses

of the fundamental mode in a single line-defect photonic crystal waveguide on

an InP membrane. Journal of Applied Physics, 92(5):2227–2234, 2002.

[41] H. Benisty, S. Olivier, C. Weisbuch, M. Agio, M. Kafesaki, C. M. Soukoulis,

M. Qiu, M. Swillo, A. Karlsson, B. Jaskorzynska, A. Talneau, J. Moosburger,

M. Kamp, A. Forchel, R. Ferrini, R. Houdre, and U. Oesterle. Models and

measurements for the transmission of submicron-width waveguide bends defined

in two-dimensional photonic crystals. IEEE Journal of Quantum Electronics,

38(7):770–785, 2002.

[42] A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, C. M. Soukoulis, and

M. Agio. Photonic-crystal ultrashort bends with improved transmission and low

reflection at 1.55 µm. Applied Physics Letters, 80(4):547–549, 2002.

[43] Marin Soljacic, Steven G. Johnson, Shanhui Fan, Mihai Ibanescu, Erich Ippen,

and J. D. Joannopoulos. Photonic-crystal slow-light enhancement of nonlinear

phase sensitivity. Journal of the Optical Society of America B, 19(9):2052–2059,

2002.

[44] M. L. M. Balistreri, H. Gersen, J. P. Korterik, L. Kuipers, and N. F. van Hulst.

Tracking femtosecond laser pulses in space and time. Science, 294:1080–1082,

2001.

165

[45] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama.

Extremely large group-velocity dispersion of line-defect waveguides in photonic

crystal slabs. Physical Review Letters, 87(25):253902, 2001.

[46] Sheng Lan, Satoshi Nishikawa, Hiroshi Ishikawa, and Osamu Wada. Engineering

photonic crystal impurity bands for waveguides, all-optical switches and optical

delay lines. IEICE Transactions on Electronics, (1):181–189, 2002.

[47] Kazuhiko Hosomi and Toshio Katsuyama. Photonic-crystal slow-light enhance-

ment of nonlinear phase sensitivity. IEEE Journal of Quantum Electronics,

38(7):825–829, 2002.

[48] Y. J. Chai, C. N. Morgan, R. V. Penty, I. H. White, T. J. Karle, and T. F. Krauss.

Propagation of optical pulse in photonic crystal waveguides. IEE Proceedings.

Optoelectronics, 151(2):109–113, 2004.

[49] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus,

and I. Kim. Two-dimensional photonic band-gap defect mode laser. Science,

284:1819–1821, 1999.

[50] Susumu Noda, Alongkarn Chutinan, and Masahiro Imada. Trapping and emis-

sion of photons by a single defect in a photonic bandgap structure. Nature,

407:604–606, 2000.

[51] Hitomichi Takano, Bong-Shik Song, Takashi Asano, and Susumu Noda. Highly

efficient multi-channel drop filter in a two-dimensional hetero photonic crystal.

Optics Express, 14(8):3491–3496, 2006.

[52] Marko Loncar, Axel Scherer, and Yueming Qiu. Photonic crystal laser sources

for chemical detection. Applied Physics Letters, 82(26):4648–4650, 2003.

[53] Mark L. Adams, Marko Loncar, Axel Scherer, and Yueming Qiu. Microfluidic

integration of porous photonic crystal nanolasers for chemical sensing. IEEE

Journal On Selected Areas In Communications, 23(7):1348–1354, 2005.

166

[54] Jelena Vuckovic, Marko Loncar, Hideo Mabuchi, and Axel Scherer. Design of

photonic crystal microcavities for cavity QED. Physical Review E, 65:016608,

2001.

[55] H. Carmichael. An Open Systems Approach to Quantum Optics. Lecture Notes

in Physics, Monographs Series, v.18. Published by Springer-Verlag (Berlin Hei-

delberg), 1993., 1993.

[56] H. J. Kimble. Strong interactions of single atoms and photons in cavity QED.

Physica Scripta, pages 127–137, 1998.

[57] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi. Quantum state transfer and

entanglement distribution among distant nodes in a quantum network. Physical

Review Letters, 78(16):3221–3224, 1997.

[58] H. Mabuchi, M. Armen, B. Lev, M. Loncar, J. Vuckovic, H. J. Kimble, J. Preskill,

M. Roukes, and A. Scherer. Quantum networks based on cavity QED. Quantum

Information and Computation, 1(Special):7–12, 2001.

[59] Benjamin Lev, Kartik Srinivasan, Paul Barclay, Oskar Painter, and Hideo

Mabuchi. Feasibility of detecting single atoms using photonic bandgap cavities.

Nanotechnology, 15:S556–S561, 2004.

[60] Paul E. Barclay, Kartik Srinivasan, Oskar Painter, Benjamin Lev, and Hideo

Mabuchi. Integration of fiber-coupled high-Q SiNx microdisks with atom chips.

Applied Physics Letters, 89:131108, 2006.

[61] Takao Aoki, Barak Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippen-

berg, K. J. Vahala, and H. J. Kimble. Observation of strong coupling between

one atom and a monolithic microresonator. Nature, 443:671–674, 2006.

[62] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble. Mea-

surement of conditional phase shifts for quantum logic. Physical Review Letters,

75:4710–4713, 1995.

167

[63] Kartik Srinivasan and Oskar Painter. Fourier space design of high-Q cavities in

standard and compressed hexagonal lattice photonic crystals. Optics Express,

11(6):579–593, 2003.

[64] Dirk Englund, Ilya Fushman, and Jelena Vuckovic. General recipe for designing

photonic crystal cavities. Optics Express, 13(16):5961–5975, 2005.

[65] Kartik Srinivasan, Paul E. Barclay, and Oskar Painter. Fabrication-tolerant high

quality factor photonic crystal microcavities. Optics Express, 12(7):1458–1463,

2004.

[66] Ioan L. Gheorma, Stephan Haas, and A. F. J. Levi. Aperiodic nanophotonic

design. Journal of Applied Physics, 95(3):1420–1426, 2004.

[67] J. M. Geremia, Jon B. Williams, and Hideo Mabuchi. An inverse-problem

approach to designing photonic crystals for cavity QED. Physical Review E,

66:066606, 2002.

[68] Martin Burger, Stanley J. Osher, and Eli Yablonovitch. Inverse problem tech-

niques for the design of photonic crystals. IEICE Transactions on Electronics,

(3):258–265, 2004.

[69] Konstantin V. Popov and Alexander V. Tikhonoravov. The inverse problem

in optics of stratified media with discontinuous parameters. Inverse Problems,

13:801–814, 1997.

[70] Habib Ammari. Uniqueness theorems for an inverse problem in a doubly periodic

structure. Inverse Problems, 11:823–833, 1995.

[71] Steven J. Cox and David C. Dobson. Maximizing band gaps in two-dimensional

photonic crystals. SIAM Journal of Applied Math, 59(6):2108–2120, 1999.

[72] Steven J. Cox and David C. Dobson. Band structure optimization of two-

dimensional photonic crystals in H-polarization. Journal of Computational

Physics, 158:214–224, 2000.

168

[73] Andreas Hakansson, Jose Sanchez-Dehesa, and Lorenzo Sanchis. Inverse design

of photonic crystal devices. IEEE Journal on Selected Areas in Communications,

23(7):1365–1371, 2005.

[74] Andreas Hakansson and Jose Sanchez-Dehesa. Inverse designed photonic crystal

de-multiplex waveguide coupler. Optics Express, 13(14):5440–5449, 2005.

[75] Jakob S. Jensen and Ole Sigmund. Systematic design of photonic crystal struc-

tures using topology optimization: Low-loss waveguide bends. Applied Physics

Letters, 84(12):2022–2024, 2004.

[76] Jakob S. Jensen and Ole Sigmund. Topology optimization of photonic crystal

structures: a high-bandwidth low-loss T-junction waveguide. Journal of the

Optical Society of America B, 22(6):1191–1198, 2005.

[77] W. R. Frei, D. A. Tortorelli, and H. T. Johnson. Topology optimization of a pho-

tonic crystal waveguide termination to maximize directional emission. Applied

Physics Letters, 86:111114, 2005.

[78] Tosio Kato. A short introduction to perturbation theory for linear operators.

Springer-Verlag, 1982.

[79] Thomas Felici and Heinz W. Engl. On shape optimization of optical waveguides

using inverse problem techniques. Inverse Problems, 17:1141–1162, 2001.

[80] Stanley J. Osher and Ronald P. Fedkiw. Level Set Methods and Dynamic Implicit

Surfaces. Springer Verlag Publishing, 2002.

[81] C. Y. Kao, S. Osher, and E. Yablonovitch. Maximizing band-gaps in two-

dimensional photonic crystals by using level set methods. Applied Physics B,

81:235–244, 2005.

[82] H. Benisty. Photonic crystals: New designs to confine light. Nature Physics,

1:9–10, October 2005.

169

[83] Don Johnson. Optimization Theory. Optimization Theory page from the Con-

nexions Project; http://cnx.org/content/m11240/latest/.

[84] James W. Cooley and John W. Tukey. An algorithm for the machine calculation

of complex Fourier series. Math. Comput., 19:297–301, 1965.

[85] Matteo Frigo and Steven G. Johnson. The design and implementation of FFTW3.

Proceedings of the IEEE, 93(2):216–231, 2005. special issue on “Program Gen-

eration, Optimization and Platform Adaptation”.

[86] R. M. Gray. Toeplitz and circulant matrices: a review. Foundations and Trends

in Communications and Information Theory, 2(3):155–239, 2006.

[87] Philippe Lalanne. Effective properties and band structures of lamellar subwave-

length crystals: Plane-wave method revisited. Physical Review B, 58(15):9801–

9807, 1998.

[88] Lifeng Li. New formulation of the Fourier modal method for crossed surface-

relief gratings. Journal of the Optical Society of America B, 14(10):2758–2767,

1997.

[89] G. H. Golub and C. F. van Loan. Matrix Computations. Johns Hopkins Univer-

sity Press, 1983.